![x INTRODUCTION
a comprehensive presentation of the many facets of stochastic resonance in various
areas of sciences (a sample will be briefly discussed in Chapter 1, Section 1.5). In
particular it does not touch computational aspects relevant in particular in high
dimensions where analytical methods alone are too complex to be of practical use
any more (for an incomplete overview of stochastic resonance from a computational
dynamics perspective see also Chapter 1, Section 1.5).
We now explain briefly our motivation and approach. The most prominent and
one of the first examples in which phenomena related to stochastic resonance were
observed is given by energy balance models of low dimensional conceptual climate
dynamics. It was employed for a qualitative explanation of glacial cycles in earth’s
history, i.e. the succession of ice and warm ages observed in paleoclimatic data,
by means of stochastic transitions between cold and warm meta-stable climates in
a dynamical model. It will be discussed in more detail in Chapter 1. The model
proposed by Nicolis [83] and Benzi et al. [6] is based on the balance between aver-
aged absorbed and emitted radiative energy and leads to a deterministic differential
equation for averaged global temperature T of the form
Ṫ(t) = b(t, T(t)).
The explicit time dependence of b captures the influence of the solar constant that
undergoes periodic fluctuations of a very small amplitude at a very low frequency.
The fluctuations are due to periodic changes of the earth’s orbital parameters (Mi-
lankovich cycles), for instance a small variation of the axial tilt that arises at a
frequency of roughly 4 × 10−4
times per year, and coincide roughly with the ob-
served frequencies of cold and warm periods. For frozen t the nonlinear function
b(t, T) describes the difference between absorbed radiative energy as a piecewise
linear function of the temperature dependent albedo function a(T) and emitted ra-
diative energy proportional to T4
due to the Stefan–Boltzmann law of black body
radiators. In the balance for relevant values of T it can be considered as negative
gradient (force) of a double well potential, for which the two well bottoms corre-
spond to stable temperature states of glacial and warm periods. The evolution of
temperature in the resulting deterministic dynamical system is analogous to the
motion of an overdamped physical particle subject to the weakly periodic force
field of the potential. Trajectories of the deterministic system relax to the stable
states of the domain of attraction in which they are started. Only the addition of
a stochastic forcing to the system allows for spontaneous transitions between the
different stable states which thus become meta-stable.
In a more general setting, we study a dynamical system in d-dimensional Eu-
clidean space perturbed by a d-dimensional Brownian motion W, i.e. we consider
the solution of the stochastic differential equation
(0.1) dXε
t = b
t
T
, Xε
t
dt +
√
ε dWt, t ≥ 0.
One of the system’s important features is that its time inhomogeneity is weak in
the sense that the drift depends on time only through a re-scaling by the time
parameter T = T(ε) which will be assumed to be exponentially large in ε. This
corresponds to the situation in Herrmann and Imkeller [50] and is motivated by
the well known Kramers–Eyring law which was mathematically underpinned by the
Freidlin–Wentzell theory of large deviations [40]. The law roughly states that the
expected time it takes for a homogeneous diffusion to leave a local attractor e.g.](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-15-320.jpg)
![INTRODUCTION xi
across a potential wall of height v
2 is given to exponential order by T(ε) = exp(v
ε ).
Hence, only in exponentially large scales of the form T(ε) = exp(μ
ε ) parametrized by
an energy parameter μ we can expect to see effects of transitions between different
domains of attraction. We remark at this place that our approach essentially differs
from the one by Berglund and Gentz [13]. If b represents a negative potential
gradient for instance, their approach would typically not only scale time by T,
but also the depths of the potential wells by a function of ε. As a consequence,
transitions even for the deterministic dynamical system become possible, and their
noise induced transitions happen on time scales of intermediate length. In contrast,
in our setting transitions between the domains of attraction of the deterministic
system are impossible, and noise induced ones are observed on very large time scales
of the order of Kramers’ time, typically as consequences of large deviations. The
function b is assumed to be one-periodic w.r.t. time, and so the system described
by (0.1) attains period T by re-scaling time in fractions of T. The deterministic
system ˙
ξt = b(s, ξt) with frozen time parameter s is supposed to have two domains of
attraction that do not depend on s ≥ 0. In the “classical” case of a drift derived from
a potential, b(t, x) = −∇xU(t, x) for some potential function U, equation (0.1) is
analogous to the overdamped motion of a Brownian particle in a d-dimensional time
inhomogeneous double-well potential. In general, trajectories of the solutions of
differential equations of this type will exhibit randomly periodic behavior, reacting
to the periodic input forcing and eventually amplifying it. The problem of optimal
tuning at large periods T consists in finding a noise amplitude ε(T) (the resonance
point) which supports this amplification effect in a best possible way. During the last
20 years, various concepts of measuring the quality of periodic tuning to provide
a criterion for optimality have been discussed and proposed in many applications
from a variety of branches of natural sciences (see Gammaitoni et al. [43] for an
overview). Its rigorous mathematical treatment was initiated only relatively late.
The first approach towards a mathematically precise understanding of stochas-
tic resonance was initiated by Freidlin [39]. To explain stochastic resonance in the
case of diffusions in potential landscapes with finitely many minima (in the more
general setting of (0.1), the potential is replaced by a quasi-potential related to the
action functional of the system), he goes as far as basic large deviations’ theory can
take. If noise intensity is ε, in the absence of periodic exterior forcing, the exponen-
tial order of times at which successive transitions between meta-stable states occur
corresponds to the work to be done against the potential gradient to leave a well
(Kramers’ time). In the presence of periodic forcing with period time scale e
μ
ε , in
the limit ε → 0 transitions between the stable states with critical transition energy
close to μ will be periodically observed. Transitions with smaller critical energy
may happen, but are negligible in the limit. Those with larger critical energy are
forbidden. In case the two local minima of the potential have depths V
2 and v
2 ,
v V , that switch periodically at time 1
2 (in scale T accordingly at time T
2 ), for
T larger than e
v
ε the diffusion will be close to the deterministic periodic function
jumping between the locations of the deepest wells. As T exceeds this exponen-
tial order, many short excursions to the wrong well during one period may occur.
They will not count on the exponential scale, but trajectories will look less and
less periodic. It therefore becomes plausible that physicists’ quality measures for
periodic tuning which always feature some maximal tuning quality of the random](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-16-320.jpg)
![xii INTRODUCTION
trajectories to the periodic input signal cannot be captured by this phenomenon of
quasi-deterministic periodicity at very large time scales.
These quality measures, studied in Pavlyukevich [86] and Imkeller and Pavlyuke-
vich [59] assess quality of tuning of the stochastic output to the periodic determin-
istic input. The concepts are mostly based on comparisons of trajectories of the
noisy system and the deterministic periodic curve describing the location of the
relevant meta-stable states, averaged with respect to the equilibrium measure (of
the diffusion as a space-time process with time component given by uniform motion
in the period interval). Again in the simple one-dimensional situation considered
above the system switches between a double well potential state U with two wells of
depths V
2 and v
2 , v V, during the first half period, and the spatially opposite one
U(·) for the second half period. If as always time is re-scaled by T, the total period
length is T, and stochastic perturbation comes from the coupling to a white noise
of intensity ε. The most important measures of quality studied are the spectral
power amplification and the related signal-to-noise ratio, both playing an eminent
role in the physical literature (see Gammaitoni et al. [43], Freund et al. [41]). They
mainly contain the mean square average in equilibrium of the Fourier component
of the solution trajectories corresponding to the input period T, normalized in dif-
ferent ways. These measures of quality are functions of ε and T, and the problem
of finding the resonance point consists in optimizing them in ε for fixed (large) T.
Due to the high complexity of original systems, when calculating the resonance
point at optimal noise intensity, physicists usually pass to an effective dynamics
description. It is given by a simple caricature of the system reducing the diffu-
sion dynamics to the pure inter well motion (see e.g. McNamara and Wiesenfeld
[74]). The reduced dynamics is represented by a continuous time two state Markov
chain with transition probabilities corresponding to the inverses of the diffusions’
Kramers’ times. One then determines the optimal tuning parameters ε(T) for large
T for the approximating Markov chains in equilibrium, a rather simple task. To
see that the Markov chain’s behavior approaches the diffusion’s in the small noise
limit, spectral theory for the infinitesimal generator is used. The latter is seen to
possess a spectral gap between the second and third eigenvalues, and therefore the
closeness of equilibrium measures can be well controlled. Surprisingly, due to the
importance of small intra well fluctuations, the tuning and resonance pattern of
the Markov chain model may differ dramatically from the resonance picture of the
diffusion. Subtle dependencies on the geometrical fine structure of the potential
function U in the minima beyond the expected curvature properties lead to quite
unexpected counterintuitive effects. For example, a subtle drag away from the other
well caused by the sign of the third derivative of U in the deep well suffices to make
the spectral power amplification curve strictly increasing in the parameter range
where the approximating Markov chain has its resonance point.
It was this lack of robustness against model reduction which motivated Her-
rmann and Imkeller [50] to look for different measures of quality of periodic tun-
ing for diffusion trajectories. These notions are designed to depend only on the
rough inter well motion of the diffusion. The measure treated in the setting of
one-dimensional diffusion processes subject to periodic forcing of small frequency
is related to the transition probability during a fixed time window of exponential
length T(ε) = exp(μ
ε ) parametrized by a free energy parameter μ according to the](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-17-320.jpg)
![INTRODUCTION xiii
Kramers–Eyring formula. The corresponding exit rate is maximized in μ to ac-
count for optimal tuning. The methods of investigation of stochastic resonance in
[50] are heavily based on comparison arguments which are not an appropriate tool
from dimension 2 on. Time inhomogeneous diffusion processes such as the ones
under consideration are compared to piecewise homogeneous diffusions by freezing
the potential’s time dependence on small intervals.
In Herrmann et al. [51] this approach is extended to the general setting of finite
dimensional diffusion processes with two meta-stable states. Since the stochastic
resonance criterion considered in [50] is based on transition times between them,
our analysis relies on a suitable notion of transition or exit time parametrized again
by the free energy parameter μ from T(ε) = exp(μ
ε ) as a natural measure of scale.
Assume now that the depths of the two equilibria of the potential in analogy to
the scenarios considered before are smooth periodic functions of time of period 1
given for one of them by v(t)
2 , and for the other one by the same function with some
phase delay (for instance by 1
2 ). Therefore, at time s the system needs energy v(s)
to leave the domain of attraction of the equilibrium. Hence an exit from this set
should occur at time
aμ = inf{t ≥ 0: v(t) ≤ μ}
in the diffusion’s natural time scale, in the time re-scaled by T(ε) thus at time
aμ · T(ε). To find a quality measure of periodic tuning depending only on the
transition dynamics, we look at the probabilities of transitions to the other domain
within a time window [(aμ − h)T(ε), (aμ + h)T(ε)] centered at aμ · T(ε) for small
h 0. If τ is the random time at which the diffusion roughly reaches the other
domain of attraction (to be precise, one has to look at first entrance times of small
neighborhoods of the corresponding equilibrium), we use the quantity (again, to be
precise, we use the worst case probability for the diffusion starting in a point of a
small neighborhood of the equilibrium of the starting domain)
Mh
(ε, μ) = P
τ ∈ [(aμ − h)T(ε), (aμ + h)T(ε)]
.
To symmetrize this quality measure with respect to switching of the equilibria, we
refine it by taking its minimum with the analogous expression for interchanged
equilibria. In order to exclude trivial or chaotic transition behavior, the scale
parameter μ has to be restricted to an interval IR of reasonable values which we
call resonance interval. With this measure of quality, the stochastic resonance point
may be determined as follows. We first fix ε and the window width parameter h 0,
and maximize Mh
(ε, μ) in μ, eventually reached for the time scale μ0(h). Then the
eventually existing limit limh→0 μ0(h) will be the resonance point.
To calculate μ0(h) for fixed positive h we use large deviations techniques. In
fact, our main result consists in an extension of the Freidlin–Wentzell large devia-
tions result to weakly time inhomogeneous dynamical systems perturbed by small
Gaussian noise which states that
lim
ε→0
ε ln
1 − Mh
(ε, μ)
= μ − v(aμ − h),
again in a form which is symmetric for switched equilibria. We show that this
asymptotic relation holds uniformly w.r.t. μ on compact subsets of IR, a fact which
enables us to perform a maximization and find μ0(h). The resulting notion of
stochastic resonance is strongly related to the notions of periodic tuning based
on interspike intervals (see [49]), which describe the probability distribution for](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-18-320.jpg)
![xiv INTRODUCTION
transitions as functions of time with exponentially decaying spikes near the integer
multiples of the forcing periods. It has the big advantage of being robust for
model reduction, i.e. the passage from the diffusion to the two state Markov chain
describing its reduced dynamics.
The techniques needed to prove this main result feature non-trivial extensions
and refinements of the fundamental large deviations theory for time homogeneous
diffusions by Freidlin–Wentzell [40]. We prove a large deviations principle for the
inhomogeneous diffusion (0.1) and further strengthen this result to get uniformity in
system parameters. Similarly to the time homogeneous case, where large deviations
theory is applied to the problem of diffusion exit culminating in a mathematically
rigorous proof of the Kramers–Eyring law, we study the problem of diffusion exit
from a domain which is carefully chosen in order to allow for a detailed analysis of
transition times. The main idea behind our analysis is that the natural time scale
is so large that re-scaling in these units essentially leads to an asymptotic freezing
of the time inhomogeneity, which has to be carefully controlled, to hook up to the
theory of large deviations of time homogeneous diffusions.
The material in the book is organized as follows. In Chapter 1 we give a de-
tailed treatment of the heuristics behind our mathematical approach, mostly in
space dimension 1. We start by giving a fairly thorough account of the paradigm
of glacial cycles which was the historical root of physical models exhibiting sto-
chastic resonance. It gives rise to the model equation of a weakly periodically
forced dynamical system with noise that can be interpreted as the motion of an
overdamped physical particle in a weakly periodically forced potential landscape
subject to noise. The heuristics of exit and transition behavior between domains of
attraction (potential wells) of such systems based on the classical large deviations
theory is explained in two steps: first for time independent potential landscapes,
then for potentials switching discontinuously between two anti-symmetric states
every half period. Freidlin’s quasi-deterministic motion is seen to not cover the
concept of optimal periodic tuning between weak periodic input and randomly am-
plified output. They determine stochastic resonance through measures of quality of
periodic tuning such as the spectral power amplification or the signal-to-noise ratio.
The latter concepts are studied first for finite state Markov chains capturing the
dynamics of the underlying diffusions reduced to the meta-stable states, and then
for the diffusions with time continuous periodic potential functions. The robustness
defect of the classical notions of resonance in passing from Markov chain to diffu-
sion is pointed out. Then alternative notions of resonance are proposed which are
based purely on the asymptotic behavior of transition times. Finally, examples of
systems exhibiting stochastic resonance features from different areas of science are
presented and briefly discussed. They document the ubiquity of the phenomenon
of stochastic resonance.
Our approach is based on concepts of large deviations. Therefore Chapter 2 is
devoted to a self-contained treatment of the theory of large deviations for randomly
perturbed dynamical systems in finite dimensions. Following a direct and elegant
approach of Baldi and Roynette [3], we describe Brownian motion in its Schauder
decomposition. It not only allows a direct approach to its regularity properties in
terms of Hölder norms on spaces of continuous functions. It also allows a derivation
of Schilder’s large deviation principle (LDP) for Brownian motion from the elemen-
tary LDP for one-dimensional Gaussian random variables. The key to this elegant](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-19-320.jpg)
![INTRODUCTION xv
and direct approach is Ciesielski’s isomorphism of normed spaces of continuous
functions with sequence spaces via Fourier representation. The proof of the LDP
for Brownian motion using these arguments is given after recalling general notions
and basic concepts about large deviations, especially addressing their construction
from exponential decay rates of probabilities of basis sets of topologies, and their
transport between different topological spaces via continuous mappings (contrac-
tion principle). Since we only consider diffusion processes with additive noise for
which Itô’s map is continuous, an appeal to the contraction principle provides the
LDP for the homogeneous diffusion processes we study. Finally, we follow Dembo
and Zeitouni [25] to derive the exit time laws due to Freidlin and Wentzell [40] for
time homogeneous diffusions from domains of attraction of underlying dynamical
systems in the small noise limit.
Chapter 3 deals with an approach to stochastic resonance for diffusions with
weakly time periodic drift and additive noise in the spirit of the associated Mar-
kovian semigroups and their spectral theory. This approach, presented in space
dimension 1, is clearly motivated by the physical notions of periodic tuning, in
particular the spectral power amplification coefficient. It describes the average
spectral component of the diffusion trajectories corresponding to the frequency of
the periodic input signal given by the drift term. We first give a rigorous account
of Freidlin’s quasi-deterministic limiting motion for potential double well diffusions
of this type. We then follow the paradigm of the physics literature, in particular
NcNamara and Wiesenfeld [74], and introduce the effective dynamics of our weakly
periodically forced double well diffusions given by reduced continuous time Markov
chains jumping between their two meta-stable equilibria. In this setting, different
notions of periodic tuning can easily be investigated. We not only consider the
physicists’ favorites, spectral power amplification and signal-to-noise ratio, but also
other reasonable concepts in which the energy carried by the Markov chain trajecto-
ries or the entropy of their invariant measures are used. Turning to diffusions with
weakly time periodic double well potentials and additive noise again, we then de-
velop an asymptotic analysis of their spectral power amplification coefficient based
on the spectral theory of their infinitesimal generators. It is based on the crucial
observation that in the case of double well potentials its spectrum has a gap be-
tween the second and third eigenvalue. Therefore we have to give the corresponding
eigenvalues and eigenfunctions a more detailed study, in particular with respect to
their asymptotic behavior in the small noise limit. Its results then enable us to give
a related small noise asymptotic expansion both of the densities of the associated
invariant measures as for the spectral power amplification coefficients. We finally
compare spectral power amplification coefficients of the Markov chains describing
the reduced dynamics and the associated diffusions, to find that in the small noise
limit they may be essentially different, caused by the small oscillations catastrophe
near the potential wells’ bottoms.
This motivates us in Chapter 4 to look for notions of periodic tuning for the so-
lution trajectories of diffusions in spaces of arbitrary finite dimension with weakly
periodic drifts and additive small noise which do not exhibit this robustness de-
fect. We aim at notions related to the maximal probabilities that the random exit
or transition times between different domains of attraction of the underlying dy-
namical systems happen in time windows parametrized by free energy parameters
on an exponential scale. For the two-state Markov chains describing the effective](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-20-320.jpg)
![CHAPTER 1
Heuristics of noise induced transitions
1.1. Energy balance models of climate dynamics
The simple concept of energy balance models stimulated research not only in
the area of conceptual climate models, but was at the cradle of a research direc-
tion in physics which subsequently took important examples from various domains
of biology, chemistry and neurology: it was one of the first examples for which
the phenomenon of stochastic resonance was used to explain the transition dynam-
ics between different stable states of physical systems. For a good overview see
Gammaitoni et al. [43] or Jung [62].
In the end of the 70’s, Nicolis [83] and Benzi et al. [5] almost simultaneously
tried stochastic resonance as a rough and qualitative explanation for the glaciation
cycles in earth’s history. They were looking for a simple mathematical model appro-
priate to explain experimental findings from deep sea core measurements according
to which the earth has seen ten glacial periods during the last million years, alter-
nating with warm ages rather regularly in periods of about 100 000 years. Mean
temperature shifts between warm age and glacial period are reported to be of the
order of 10 K, and relaxation times, i.e. transition times between two relatively
stable mean temperatures as rather short, of the order of only 100 years. Math-
ematically, their explanation was based on an equation stating the global energy
balance in terms of the average temperature T(t), where the global average is taken
meridionally (i.e. over all latitudes), zonally (i.e. over all longitudes), and annually
around time t. The global radiative power change at time t is equated to the differ-
ence between incoming solar (short wave) radiative power Rin and outgoing (long
wave) radiative power Rout.
The power Rin is proportional to the global average of the solar constant Q(t)
at time t. To model the periodicity in the glaciation cycles, one assumes that Q
undergoes periodic variations due to one of the so-called Milankovich cycles, based
on periodic perturbations of the earth’s orbit around the sun. Two of the most
prominent cycles are due to a small periodic variation between 22.1 and 24.5 degrees
of the angle of inclination (obliquity) of the earth’s rotation axis with respect to
its plane of rotation, and a very small periodic change of only about 0.1 percent of
the eccentricity, i.e. the deviation from a circular shape, of the earth’s trajectory
around the sun. The obliquity cycle has a duration of about 41 000 years, while
the eccentricity cycle corresponds to the 100 000 years observed in the temperature
proxies from deep sea core measurements mentioned above. They are caused by
gravitational influences of other planets of our solar system. In formulas, Q was
assumed to be of the form
Q(t) = Q0 + b sin ωt,
with some constants Q0, b and a frequency ω = 10−5
[1
y ].
1](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-22-320.jpg)
![4 1. HEURISTICS
0 2 · 105
3 · 105
4 · 105
105
time
T3(t)
T1(t)
Figure 1.5. Unrealistic equilibrium temperatures T1(t) and T3(t)
for large fluctuation amplitude b.
vanishes completely, and transitions are still impossible, unless one is willing to
accept discontinuous behavior (Figure 1.5).
For this reason, Nicolis [83] and Benzi et al. [5] proposed to add a noise term
in (1.1). Despite the fact that then negative temperatures become possible, they
worked with the equation
(1.2) cṪε
(t) = Q(t)
1 − a(Tε
(t))
− γ Tε
(t)4
+
√
ε Ẇt,
ε 0, where Ẇ is a white noise. In passing to (1.2), stable equilibria of the
deterministic system become — approximately at least — meta-stable states of the
stochastic system. And more importantly, the unbounded noise process W makes
spontaneous transitions (tunneling) between the meta-stable states T1(t) and T3(t)
possible. In fact, the random hopping between the meta-stable states immediately
exhibits two features which make the model based on (1.2) much more attractive for
a qualitative explanation of glaciation cycles: a) the transitions between T1 and T3
allow for far more realistic temperature shifts, b) relaxation times are random, but
very short compared to the periods the process solving (1.2) spends in the stable
states themselves.
But now a new problem arises, which actually provided the name stochastic
resonance.
If, seen on the scale of the period of Q, ε is too small, the solution may be
trapped in one of the states T1 or T3. By the periodic variation of Q, there are
well defined periodically returning time intervals during which T1(t) is the more
probable state, while T3(t) takes this role for the rest of the time. So if ε is small,
the process, initially in T1, may for example fail to leave this state during a whole
period while the other one is more probable. The solution trajectory may then look
as in Figure 1.6.
If, on the other hand, ε is too large, the big random fluctuation may lead to
eventual excursions from the actually more probable equilibrium during its domina-
tion period to the other one. The trajectory then typically looks like in Figure 1.7.
In both cases it will be hard to speak of a random periodic curve. Good tuning
with the periodic forcing by Q is destroyed by a random mechanism being too slow
or too fast to follow. It turned out in numerous simulations in a number of similar
systems that there is, however, an optimal parameter value ε for which the solution](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-25-320.jpg)
![6 1. HEURISTICS
The optimally tuned system is then said to be in stochastic resonance. Nicolis
[83] and Benzi et al. [5], by tuning the noise parameter ε to appropriate values, were
able to give qualitative explanations for glaciation cycles based on this phenomenon.
Stochastic resonance proved to be relevant in other elementary climate models
than the energy balance models considered so far. In Penland et al. [87], Wang
et al. [107, 106], a two-dimensional stochastic model for a qualitative explanation
of the ENSO (El Niño Southern Oscillation) phenomenon also leads to stochastic
resonance effects: for certain parameter ranges the model exhibits random tuned
transitions between two stable sea surface temperatures. New evidence for the pres-
ence of stochastic resonance phenomena in paleo-climatic time series was added by
Ganopolski and Rahmstorf [45]. Their paper interprets the GRIP ice core record
representing temperature proxies from the Greenland glacier that extend over a
period of roughly 90 000 years, and showing the fine structure of the temperature
record of the last glacial period. The time series shows about 20 intermediate warm-
ings during the last glacial period commonly known under the name of Dansgaard–
Oeschger events. These events are clearly marked by rapid spontaneous increases
of temperature by about 6K followed by slower coolings to return to the initial
basic cold age temperature. It was noted in [45] that a histogram of the number of
Dansgaard–Oeschger events with a duration of k · 1480 years, with k = 1, 2, 3, . . .
exhibits the typical shape of a stochastic resonance spike train consistent for in-
stance with the results of Herrmann and Imkeller [49] for Markov chains describing
the effective diffusion dynamics, or Berglund and Gentz [13] for diffusion processes
with periodic forcing.
1.2. Heuristics of our mathematical approach
The rigorous mathematical elaboration of the concept of stochastic resonance
is the main objective of this book. We start its mathematically sound treatment
by giving a heuristical outline of the main stream of ideas and arguments based on
the methods of large deviations for random dynamical systems in the framework of
the Freidlin–Wentzell theory. Freidlin [39] is able to formulate Kramers’ [65] very
old seminal approach mathematically rigorously in a very general setting, and this
way provides a lower estimate for the good tuning (see also the numerical results
by Milstein and Tretyakov [77]). To obtain an upper estimate, we finally argue by
embedding time discrete Markov chains into the diffusion processes that describe
the effective dynamics of noise induced transitions. Optimal tuning results obtained
for the Markov chains will then be transferred to the original diffusion processes.
To describe the idea of our approach, let us briefly return to our favorite ex-
ample explained in the preceding section. Recall that the function
f(t, T) = Q(t)
1 − a(T)
− γ T4
, T, t ≥ 0,
describes a multiple of Rin − Rout, and its very slow periodicity in t is initiated
by the assumption on the solar constant Q(t) = Q0 + b sin(ωt). Let us compare
this quantity, sketched in Figure 1.9 schematically for two times, say t1, t2 such
that Q takes its minimum at t1 and its maximum at t2. The graph of f moves
periodically between the two extreme positions. Note that in the one-dimensional
situation considered, f(t, ·) can be seen as the negative gradient of a potential
function U(t, ·) which depends periodically on time t.](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-27-320.jpg)
![8 1. HEURISTICS
The variable period of the input will be denoted by some positive number T. We
therefore consider the stochastic differential equation
(1.3)
d
dt
Xε
t = f
t
T
, Xε
t
+
√
ε Ẇt,
with a one-dimensional Wiener process W (white noise Ẇ). We may circumscribe
a more mathematical concept of stochastic resonance like this: given T (ω = 1
T ),
find the parameter ε = ε(T) such that Xε
is optimally tuned with the periodic
input f( t
T , ·). We pose the problem in the following (almost equivalent) way: given
ε 0, find the good scale T = T(ε) such that optimal tuning of Xε
with the
periodic input is given, at least in the limit ε → 0.
1.2.1. Random motion of a strongly damped Brownian particle. The
analogy with the motion of a physical particle in a periodically changing double
well potential landscape alluded to in (1.3) (see also Mazo [72] and Schweitzer [97])
motivates us to pause for a moment and give it a little more thought. As in the
previous section, let us concentrate on a one-dimensional setting, remarking that
our treatment easily generalizes to a finite-dimensional setting. Due to Newton’s
law, the motion of a particle is governed by the impact of all forces acting on it.
Let us denote F the sum of these forces, m the mass, x the space coordinate and
v the velocity of the particle. Then
mv̇ = F.
Let us first assume the potential to be turned off. In their pioneering work at the
turn of the twentieth century, Marian Smoluchowski and Paul Langevin introduced
stochastic concepts to describe the Brownian particle motion by claiming that at
time t
F(t) = −γv(t) +
2kTγẆt.
The first term results from friction γ and is velocity dependent. An additional sto-
chastic force represents random interactions between Brownian particles and their
simple molecular random environment. The white noise Ẇ (the formal derivative
of a Wiener process) plays the crucial role. The diffusion coefficient (standard
deviation of the random impact) is composed of Boltzmann’s constant k, friction
and environmental temperature T. It satisfies the condition of the fluctuation-
dissipation theorem expressing the balance of energy loss due to friction and energy
gain resulting from noise. The equation of motion becomes
⎧
⎨
⎩
ẋ(t) = v(t),
v̇(t) = −
γ
m
v(t) +
√
2kTγ
m
Ẇt.
In equilibrium, the stationary Ornstein–Uhlenbeck process provides its solution:
v(t) = v(0) e− γ
m t
+
√
2kTγ
m
t
0
e− γ
m (t−s)
dWs.
The ratio β := γ
m determines the dynamic behavior. Let us focus on the over-
damped situation with large friction and very small mass. Then for t 1
β = τ
(relaxation time), the first term in the expression for velocity can be neglected,
while the stochastic integral represents a Gaussian process. By integrating, we](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-29-320.jpg)
![1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 9
obtain in the over-damped limit (β → ∞) that v and thus x is Gaussian with
almost constant mean
m(t) = x(0) +
1 − e−βt
β
v(0) ≈ x(0)
and covariance close to the covariance of white noise, see Nelson [82]:
K(s, t) =
2kT
γ
min(s, t) +
kT
γβ
− 2 + 2e−βt
+ 2e−βs
− e−β|t−s|
− e−β(t+s)
≈
2kT
γ
min(s, t), s, t ≥ 0.
Hence the time-dependent change of the velocity of the Brownian particle can be
neglected, the velocity rapidly converges to thermal equilibrium (v̇ ≈ 0), while the
spatial coordinate remains far from it. In the so-called adiabatic transformation,
the evolution of the particle’s position is thus given by the transformed Langevin
equation
ẋ(t) =
√
2kT
γ
Ẇt.
Let us next suppose that we face a Brownian particle in an external field of force,
associated with a potential U(t, x), t ≥ 0, x ∈ R. This then leads to the Langevin
equation ⎧
⎨
⎩
ẋ(t) = v(t),
mv̇(t) = −γ v(t) −
∂U
∂x
(t, x(t)) +
2kTγ Ẇt.
In the over-damped limit, after relaxation time, the adiabatic elimination of the
fast variables (see Gardiner [46]) then leads to an equation similar to the one
encountered in the previous section, namely
ẋ(t) = −
1
γ
∂U
∂x
(t, x(t)) +
√
2kT
γ
Ẇt.
1.2.2. Time independent potential. We now continue discussing the heuris-
tics of stochastic resonance for systems described by equations of the type encoun-
tered in the previous two sections. To motivate the link to the theory of large
deviations, we first study the case in which U(t, ·) is given by some time inde-
pendent potential function U for all t. Following Freidlin and Wentzell [40], the
description of the asymptotics contained in the large deviations principle requires
the crucial notion of action functional. It is defined for T 0 and absolutely
continuous functions ϕ: [0, T] → R with derivative ϕ̇ by
S0T (ϕ) =
1
2
T
0
ϕ̇s −
−
∂
∂x
U
(ϕs)
2
ds.
By means of the action functional we can define the quasipotential function
V (x, y) = inf{S0T (ϕ): ϕ0 = x, ϕT = y, T 0},
for x, y ∈ R. Intuitively, V (x, y) describes the minimal work to be done in the
potential landscape given by U to pass from x to y. Keeping this in mind, the
relationship between U and V is easy to understand (for a more formal argument
see Chapter 3). If x and y are in the same potential well, we have
(1.4) V (x, y) = 2(U(y) − U(x))+
,](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-30-320.jpg)
![10 1. HEURISTICS
where a+
= a ∨ 0 = max{a, 0} denotes the positive part of a real number a. In
particular, if U(y) U(x), then V (x, y) = 0, i.e. going downhill in the landscape
does not require work. If, however, x and y are in different potential wells, we have
(recall U(0) = 0)
(1.5) V (x, y) = −2U(x).
This equation reflects the fact that the minimal work to do to pass to y consists in
reaching the saddle 0, since then one can just go downhill.
Rudiments of the following arguments can also be found in the explanation of
stochastic resonance by McNamara and Wiesenfeld [74]. The main ingredient is
the exit time law by Freidlin and Wentzell [40] (see also Eyring [37], Kramers [65]
and Bovier et al. [14]). For y ∈ R, ε 0 the first time y is visited is defined by
τε
y = inf{t ≥ 0: Xε
t = y}.
If Px denotes the law of the diffusion (Xε
t )t≥0 started at x ∈ R, the exit time law
states that for any δ 0, x ∈ R we have
(1.6) Px
e
V (x,y)−δ
ε ≤ τε
y ≤ e
V (x,y)+δ
ε
→ 1
as ε → 0.
In other words, in the limit ε → 0, the process started at x takes approximately
time exp(V (x,y)
ε ) to reach y, or more roughly
ε ln τε
y
∼
= V (x, y)
as ε → 0. As a consequence, one finds that as ε → 0, on time scales T(ε) at least
as long as exp(V (x,y)
ε ) or such that
ε ln T(ε) V (x, y),
we may expect with Px-probability close to 1 that the process Xε
tT (ε) has reached
y by time 1. Remembering (1.4) and (1.5) one obtains the following statement
formulated much more generally by Freidlin. Suppose
(1.7) lim
ε→0
ε ln T(ε) 2 max{−U(−1), −U(1)},
and U(−1) U(1). Then the Lebesgue measure of the set
(1.8) t ∈ [0, 1]: |Xε
tT (ε) − (−1)| δ
tends to 0 in Px-probability as ε → 0, for any δ 0.
In other words, the process Xε
, run in a time scale T(ε) large enough, will
spend most of the time in the deeper potential well. Excursions to the other well
are exponentially negligible on this scale, as ε → 0. The picture is roughly as
deployed in Figure 1.11.
1.2.3. Periodic step potentials and quasi-deterministic motion. As a
rough approximation of temporally continuously varying potential functions we may
consider periodic step function potentials such as
(1.9) U(t, ·) =
U1(·), t ∈ [k, k + 1
2 ),
U2(·), t ∈ [k + 1
2 , k + 1), k ∈ N0.
We assume that both U1 and U2 are of the type described above, that U1(x) =
U2(−x), x ∈ R, and that U1 has a well of depth V
2 at −1, and a well of depth v
2 at](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-31-320.jpg)
![1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 11
Figure 1.11. Solution trajectory of the diffusion Xε
tT (ε) in the
time independent double-well potential U.
1, with V v (and U2 wells with respectively opposite roles). Let us briefly point
out the main features of the transition times for periodic step potentials described
in (1.6). According to (1.6) the exponential rate of the transition time from −1 to
1 in U1 in the small noise limit is asymptotically given by exp(V
ε ), as long as the
time scale T of the diffusion allows no switching of the potential states before, i.e.
as long as T = T(ε) exp(V
ε ). Accordingly, the transition time from 1 to −1 in
U1 is given by exp(v
ε ), as long as T = T(ε) exp(v
ε ). Similar statements hold for
transitions between states of U2. It is therefore also plausible that (1.8) generalizes
to the following statement due to Freidlin [39, Theorem 2].
Suppose
(1.10) lim
ε→0
ε ln T(ε) V.
Define
φ(t) =
−1, t ∈ [k, k + 1
2 ),
1, t ∈ [k + 1
2 , k + 1), k ∈ N0.
Then the Lebesgue measure of the set
(1.11) t ∈ [0, 1]: |Xε
t T (ε) − φ(t)| δ
tends to 0 as ε → 0 in Px-probability, for any δ 0, x ∈ R.
Again, this just means that the process Xε
, run in a time scale T(ε) large
enough, will spend most of the time in the minimum of the deepest potential well
which is given by the time periodic function φ. Excursions to the other well are
exponentially negligible on this scale, as ε → 0. The picture is typically the one
depicted in Figure 1.12.
1.2.4. Periodic potentials and quasi-deterministic motion. Since the
function φ appearing in the previous theorem is already discontinuous, it is plausible
that the step function potential is in fact a reasonable approximation of the general
case of continuously and (slowly) periodically changing potential functions. It is
intuitively clear how the result has to be generalized to this situation. We just have
to replace the periodic step potentials by potentials frozen along a partition of the
period interval on the potential state taken at its starting point, and finally let the
mesh of the partition tend to 0. To continue the discussion in the spirit of the
previous section and with the idea of instantaneously frozen potential states, we](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-32-320.jpg)
![12 1. HEURISTICS
Figure 1.12. Solution trajectory of the diffusion Xε
tT (ε) in the
double-well periodic step potential.
have to explain the asymptotics of the minimal time a Brownian particle needs to
exit from the (frozen) starting well, say the left one. Freezing the potential at some
time s, the asymptotics of its exit time is derived from the classical large deviation
theory of randomly perturbed dynamical systems, see Freidlin and Wentzell [40].
Let us assume that U is locally Lipschitz continuous. We recall that for any t ≥ 0
the potential U(t, ·) has its minima at −1 and 1, separated by the saddle point
0. The law of the first exit time τε
1 = inf{t ≥ 0: Xε
t 0} is described by some
particular functional related to large deviation. For t 0, we introduce the action
functional on the space of real valued continuous functions C([0, t], R) on [0, t] by
Ss
t (ϕ) =
⎧
⎨
⎩
1
2
t
0
ϕ̇u +
∂
∂x
U(s, ϕu)
2
du if ϕ is absolutely continuous,
+∞ otherwise,
which is non-negative and vanishes on the set of solutions of the ordinary differential
equation ϕ̇ = − ∂
∂x U(s, ϕ). Let x and y be real numbers. With respect to the
(frozen) action functional, we define the (frozen) quasipotential
Vs(x, y) = inf{Ss
t (ϕ): ϕ ∈ C([0, t], R), ϕ0 = x, ϕt = y, t ≥ 0}
which represents the minimal work the diffusion with a potential frozen at time
s and starting in x has to do in order to reach y. To switch wells, the Brownian
particle starting in the left well’s bottom −1 has to overcome the barrier. So we let
V s = Vs(−1, 0).
This minimal work needed to exit from the left well can be computed explicitly,
and is equal to twice its depth at time s. The asymptotic behavior of the exit time
is expressed by
lim
ε→0
ε ln Exτε
1 = V s
or in generalization of (1.6)
lim
ε→0
Px
e
V s−δ
ε τε
1 e
V s+δ
ε
= 1 for any δ 0 and x 0.
Let us now assume that the left well is the deeper one at time s. If the Brownian
particle has enough time to cross the barrier, i.e. if T(ε) e
V s
ε , then, generalizing](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-33-320.jpg)
![1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 13
(1.8), Freidlin in [39, Theorem 1] proves that independently of the starting point
x it should stay near −1 in the following sense. The Lebesgue measure of the set
t ∈ [0, 1]: |Xε
tT (ε) − (−1)| δ
converges to 0 in probability as ε → 0. If T(ε) e
V s
ε , the time left is not long
enough for any crossing: the particle, starting at x, stays in the starting well, near
the stable equilibrium point. In other words, the Lebesgue measure of the set
t ∈ [0, 1]: |Xε
tT (ε) − (−I(−∞,0)(Xε
tT (ε)) + I[0,∞)(Xε
tT (ε)))| δ
converges to 0 in the small noise limit. This observation is at the basis of Freidlin’s
law of quasi-deterministic periodic motion discussed in the subsequent section. The
lesson it teaches is this: to observe switching of the position to the energetically
most favorable well, T(ε) should be larger than some critical level e
λ
ε , where λ =
infs≥0 V s. Measuring time in exponential scales by μ through the equation T(ε) =
e
μ
ε , the condition translates into μ λ. Continuing the reasoning of the preceding
subsection, if this condition is satisfied, we may define a periodic function φ denoting
the deepest well position in dependence on t. Then, in generalization of (1.11), the
process Xε
will spend most of the time, measured by Lebesgue’s measure, near φ
for small ε.
1.2.5. Quality of periodic tuning and reduced motion. Do the mani-
festations of quasi-deterministic motion in instantaneously frozen potentials just
discussed explain stochastic resonance? The problem is obvious. They just give
lower bounds for the scale T(ε) = e
μ
ε for which noise strength ε leads to random
switches between the most probable potential wells near the (periodic) deterministic
times when the role of the deepest well switches. But if μ is too big, occasional
excursions into the higher well will destroy a truly periodic tuning with the po-
tential (see Figure 1.12). Just the duration of the excursions, being exponentially
smaller than the periods of dwelling in the deeper well, will not be noticed by
the residence time criteria discussed. We therefore also need an upper bound for
possible scales. In order to find this optimal tuning scale μR λ, we first have
to measure goodness of periodic tuning of the trajectories of the solution. In the
huge physics literature on stochastic resonance, two families of criteria can be dis-
tinguished. The first one is based on invariant measures and spectral properties of
the infinitesimal generator associated with the diffusion Xε
. Now, Xε
is not time
autonomous and consequently does not admit invariant measures. By taking into
account deterministic motion of time in the interval of periodicity and considering
the time autonomous process Zε
t = (t mod T(ε), Xε
t ), t ≥ 0, we obtain a Markov
process with an invariant measure νε
t (x) dt dx. In particular, for t ≥ 0 the law of
Xε
t ∼ νε
t (x) dx and the law of Xε
t+T (ε) ∼ νε
t+T (ε)(x) dx, under this measure are the
same for all t ≥ 0. Let us present the most important notions of quality of tuning
(see Jung [62], or Gammaitoni et al. [43]):
• the spectral power amplification (SPA) which plays an eminent role in
the physics literature and describes the energy carried by the spectral
component of the averaged trajectories of Xε
corresponding to the period
of the signal:
ηX
(ε, T) =
1
0
EνXε
sT · e2πis
ds
2
, ε 0, T 0.](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-34-320.jpg)
![14 1. HEURISTICS
• the total energy of the averaged trajectories
EnX
(ε, T) =
1
0
EνXsT
2
ds, ε 0, T 0.
The second family of criteria is more probabilistic. It refers to quality measures
purely based on the location of transition times between domains of attraction of
the local minima, and residence time distributions measuring the time spent in one
well between two transitions, or interspike times. This family, to be discussed in
more detail in Section 1.4 below is certainly less popular in the physics community.
As will turn out later, these physical notions of quality of periodic tuning
of random trajectories exhibit one important drawback: they are not robust with
respect to model resolution. It is here that an important concept of model reduction
enters the stage. It is based on the conjecture that the effective dynamical properties
of periodically forced diffusion processes as given by (1.3) can be traced back to
finite state Markov chains periodically hopping between the stable equilibria of
the potential function underlying the diffusion, for which the smallness parameter
of the noise intensity is simply reflected in the transition matrix. These Markov
chains should be designed to capture the essential information about the inter-well
dynamics of the diffusion, while intra-well small fluctuations of the diffusion near
the potential minima are neglected. Investigating goodness of tuning according to
the different physical measures of quality makes sense both for the Markov chains
as for the diffusions. This idea of model reduction was captured and followed in the
physics literature in Eckmann and Thomas [32], McNamara and Wiesenfeld [74],
and Nicolis [83]. In fact, theoretical work on the concept of stochastic resonance
in the physics literature is based on the model reduction approach, see the surveys
Anishchenko et al. [1], Gammaitoni et al. [43, 44], Moss et al. [79], and Wellens
et al. [108].
As we shall see in Chapter 3, the optimal tuning relations between ε and T
do not necessarily agree for Markov chains and diffusions. Even in the small noise
limit discrepancies may persist that are caused by very subtle geometric properties
of the potential function. It is our goal to present a notion of quality of periodic
tuning which possesses this robustness property when passing from the Markov
chains capturing the effective dynamics to the original diffusions. For this reason
we shall study the different physical notions of quality of tuning first in the context
of typical finite state Markov chains with periodically forced transition matrices.
1.3. Markov chains for the effective dynamics and the physical
paradigm of spectral power amplification
To keep this heuristic exposition of the main ideas of our mathematical ap-
proach as simple as possible, besides allowing only two states for our Markov chain
that play the role of the stable equilibria of the potential −1 and 1, let us also dis-
cretize time. We continue to assume as in the discussion of periodically switching
potential states above that U1(−1) = U2(1) = −V
2 , and U1(1) = U2(−1) = −v
2 .
In a setting better adapted to our continuous time diffusion processes, in Chapter
3 time continuous Markov chains switching between two states will capture the
effective diffusion dynamics. Hence, we follow here Pavlyukevich [86] and Imkeller
and Pavlyukevich [58] and shall assume in this section that the parameter T in
our model describing the period length, is an even integer. So for T ∈ 2N, ε 0,](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-35-320.jpg)
![18 1. HEURISTICS
the amplification coefficient ηY
(ε, T) as a function of ε. So the stochastic resonance
point is marked by the maximum of the spectral power amplification coefficient as
a function of ε. To calculate it, substitute e−1/ε
= x, and differentiate the explicit
formula (1.14). The resulting relationship between period length T(ε) and noise
intensity ε marking the stochastic resonance point can be recast in the formula
T(ε) ∼
=
1
2π
pq
V − v
v
exp
V + v
2ε
.
The maximal value of spectral power amplification is given by
lim
ε→0
ηY
(ε, T(ε)) =
4
π2
0 ε
1 2
ηY
(ε, T)
4
π2
Figure 1.13. The coefficient of the spectral power amplification
ε → ηY
(ε, T) for p = q = 0.5, V = 2, v = 1, T = 10 000.
We also see that the spectral power amplification as a measure of quality of sto-
chastic resonance allows to distinguish a unique time scale, and find its exponential
rate V +v
2 together with the pre-exponential factor. The optimal exponential rate
is therefore given by the arithmetic mean of the two potential barriers marked by
the deep and shallow well of our double well potential. This basic relationship will
appear repeatedly at different stages of our mathematical elaboration of concepts
of optimal tuning.
We may summarize our findings so far for discrete Markov chains that capture
the effective dynamics of the potential diffusions which are our main subject of in-
terest. Following the physics literature (e.g. Gammaitoni et al. [43] and McNamara
and Wiesenfeld [74]) we understand stochastic resonance as optimal spectral power
amplification. The closely related notion of signal-to-noise ratio and other reason-
able concepts based on quality measures such as the relative entropy of invariant
laws are discussed for Markov chains in Chapter 3 (see Section 3.2). The spectral
power amplification coefficient measures the power carried by the expected Fourier
coefficient in equilibrium of the Markov chain switching between the stable equilib-
ria of the potential landscape of the diffusion which corresponds to the frequency
of the underlying periodic deterministic signal.
1.4. Diffusions with continuously varying potentials
The concept of spectral power amplification is readily extended to Markov
chains in continuous time, still designed to capture the effective diffusion dynamics](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-39-320.jpg)
![1.4. DIFFUSIONS WITH CONTINUOUSLY VARYING POTENTIALS 19
in higher dimensions, as well as to potential diffusions themselves. This will be done
in detail in Chapter 3. However, it will turn out that diffusions and their reduced
dynamics Markov chains are not as similar as expected. Indeed, in a reasonably
large time window around the resonance point for Y ε
, the tuning picture of the
spectral power amplification for the diffusion is different. Under weak regularity
conditions on the potential, it exhibits strict monotonicity in the window. Hence
optimal tuning points for diffusion and Markov chain differ essentially. In other
words, the diffusion’s SPA tuning behavior is not robust for passage to the reduced
model (see Chapter 3, subsection 3.4.4). This strange deficiency is difficult to
explain. The main reason of this subtle effect appears to be that the diffusive
nature of the Brownian particle is neglected in the reduced model. In order to
point out this feature, we may compute the SPA coefficient of g(Xε
) where g is
a particular function designed to cut out the small fluctuations of the diffusion in
the neighborhood of the bottoms of the wells, by identifying all states there. So
g(x) = −1 (resp. 1) in some neighborhood of −1 (resp. 1) and otherwise g is the
identity. This results in
η̃X
(ε, T) =
1
0
Eνg(Xε
sT ) e2πis
ds
2
, ε 0, T ≥ 0.
In the small noise limit this quality function admits a local maximum close to the
resonance point of the reduced model: the growth rate of Topt(ε) is also given by
the arithmetic mean of the wells’ depths. So the lack of robustness seems to be due
to the small fluctuations of the particle in the wells’ bottoms.
In any case, this clearly calls for other quality measures to be used to transfer
properties of the reduced model to the original one. Our discussion indicates that
due to their emphasis on the pure transition dynamics, a second more probabilis-
tic family of quality measures should be used. This will be made mathematically
rigorous in Chapter 4. The family is composed of quality measures based on tran-
sition times between the domains of attraction of the local minima, residence times
distributions measuring the time spent in one well between two transitions, or inter-
spike times. To explain its main features there is no need to restrict to landscapes
frozen in time independent potential states on half period intervals. So from now
on the potential U(t, x) is a continuous function in (t, x). For simplicity — remain-
ing in the one-dimensional case — we further suppose that its local minima are
given by ±1, and its only saddle point by 0, independently of time. So the only
meta-stable states on the time axis are ±1. Let us denote by v−(t)
2 (resp. v+(t)
2 the
depth of the left (resp. right) well. These function are continuous and 1-periodic.
We shall assume that they are strictly monotonous between their global extrema.
Let us now consider the motion of the Brownian particle in this landscape. As
in the preceding case, according to Freidlin’s law of quasi-deterministic motion its
trajectory gets close to the global minimum, if the period is large enough. The
exponential rate of the period should be large enough to permit transitions: if
T(ε) = eμ/ε
with μ ≥ maxi=± supt≥0 vi(t) meaning that μ is larger than the max-
imal work needed to cross the barrier, then the particle often switches between
the two wells and should stay close to the deepest position in the landscape. By
defining φ(t) = 2I{v+(t)v−(t)} − 1, in the small noise limit the Lebesgue measure
of the set
{t ∈ [0, 1]: |XtT − φ(t)| δ}](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-40-320.jpg)
![20 1. HEURISTICS
converges to 0 in probability for any δ 0. But in this case many transitions occur
in practice, and the trajectory looks chaotical instead of periodic. So we have to
choose smaller periods even if we cannot assure that the particle stays close to the
global minimum since it needs some time to cross the barrier. Let us study the
transition times. For this we assume that the starting point is −1 corresponding
to the bottom of the deepest well. If the depth of the well is always larger than
μ = ε ln T(ε), then the particle does not have enough time during one period to
climb the barrier and should therefore stay in the starting well. On the contrary if
the depth of the starting well becomes smaller than μ, the transition can and will
happen. More precisely, for μ ∈ (inft≥0 v−(t), supt≥0 v−(t)) we define
a−
μ (s) = inf{t ≥ s: v−(t) ≤ μ}.
The first transition time from −1 to 1 denoted τ+ has the following asymptotic
behavior in the small noise limit: τ+/T(ε) → a−
μ (0). The second transition which
lets the particle return to the starting well will appear near the deterministic time
a+
μ (a−
μ (s))T(ε). The definitions of the coefficients a−
μ and a+
μ are similar, the depth
of the left well just being replaced by that of the right well. In order to observe
periodic behavior of the trajectory, the particle has to stay a little time in the right
well before going back. This will happen under the assumption v+(aμ(0)) μ,
that is, the right well is the deepest one at the transition time. In fact we can then
define the resonance interval IR, the set of all values μ such that the trajectories
look periodic in the small noise limit:
IR =
max
i=±
inf
t≥0
vi(t), inf
t≥0
max
i=±
vi(t)
.
On this interval trajectories approach some deterministic periodic limit. We now
outline the construction of a quality measure that is based on these observations, to
be optimized in order to obtain stochastic resonance as the best possible response
to periodic forcing. The measure we consider is based on the probability that a
random transition of the diffusion happens during a small time window around the
limiting deterministic transition time. Recall the transition times τε
±1 of Xε
to ±1.
For h 0, ε 0, T ≥ 0 let
Mh
(ε, T) = min
i=±
Pi
τε
∓1
T(ε)
∈ [ai
μ − h, ai
μ + h]
.
In the small noise limit, this quality measure tends to 1 and optimal tuning can be
obtained due to its asymptotic behavior described by the formula
lim
ε→0
ε ln(1 − Mh
(ε, T)) = max
i=±
{μ − vi(ai
μ − h)}
for μ ∈ IR, uniformly on each compact subset. This property results from classical
large deviation techniques applied to an approximation of the diffusion which is
supposed to be locally time homogeneous, and will be derived in Chapter 4. Now
we minimize the term on the left hand side in the preceding equality. In fact, if
the window length 2h is small then μ − vi(ai
μ − h) ≈ 2hv
i(ai
μ) since vi(ai
μ) = μ by
definition. The value v
i(ai
μ) is of course negative. Thus the position in which its
absolute value is maximal should be identified. At this position the depth of the
starting well drops most rapidly below the level μ.
It is clear that for h small the eventually existing global minimizer μR(h) is a
good candidate for the resonance point. To get rid of the dependence on h, we shall
consider the limit of μR(h) as h → 0 denote by μR. This limit, if it exists, is called](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-41-320.jpg)
![1.5. STOCHASTIC RESONANCE IN MODELS FROM ELECTRONICS TO BIOLOGY 21
the resonance point of the diffusion with time periodic landscape U. Let us note
that for v−(t) = V +v
4 + V −v
4 cos(2πt) and v+(t) = v−(t + π), which corresponds
to the case of periodically switching wells’ depths between v
2 to V
2 as in the frozen
landscape case described above. Then the optimal tuning is T(ε) = exp(μR
ε ) with
μR = v+V
2 . This optimal rate is equivalent to the optimal rate given by the SPA
coefficient.
The big advantage of the quality measure based on the transition times is its
robustness. Let us therefore consider the reduced model consisting in a two-state
Markov chain with the infinitesimal generator
Q(t) =
−ϕ(t) ϕ(t)
ψ(t) −ψ(t)
,
where ϕ(t) = exp(−v−(t/T )
ε ) and ψ(t) = exp(−v+(t/T )
ε ). The distribution of transi-
tion times of this Markov chain is well known (see Chapter 4) and, divided by the
period length, converges to ai
μ. The reduced dynamics of the diffusion is captured
by the Markov chain, and the optimization of the quality measure Mh
(ε, T) for the
Markov chain and the diffusion leads to the same resonance points.
Our investigation focuses essentially on two criteria: one concerning the family
of spectral measures, especially the spectral power amplification coefficient, and the
other one dealing with transitions between the local minima of the potential. Many
other criteria for optimal tuning between weak periodic signals in dynamical systems
and stochastic response can be employed (see Chapter 3). The relation between long
deterministic periods and noise intensity usually is expressed in exponential form
T(ε) = exp(μ
ε ), since the particle needs exponentially large times to cross the barrier
separating the wells. This approach relies on the basic assumption that the barrier
height is bounded below uniformly in time. This assumption which seems natural in
the simple energy balance model of climate dynamics may be questionable in other
situations. If the barrier height becomes small periodically on a scale related to the
noise intensity, the Brownian particle does not need to wait an exponentially long
time to climb it. In this scaling trajectories may appear periodic in the small noise
limit. The modulation is assumed to be slow, but the time dependence does not
have to be assumed exponentially slow in the noise intensity. In a series of papers
[8, 9, 10, 11, 12] and in a monograph [13], Berglund and Gentz study the case in
which the barrier between the wells becomes low twice per period: at time zero the
right-hand well becomes almost flat and at the same time the bottom of the well
and the saddle approach each other; half a period later, the scenario with the roles
of the wells switched occurs. Even in this situation, there is a threshold value for
the noise intensity under which transitions are unlikely and, above this threshold,
trajectories typically exhibit two transitions per period. In this particular situation,
optimal tuning can be described in terms of the concentration of sample paths in
small space-time sets.
1.5. Stochastic resonance in models from electronics to biology
As described in the preceding sections, the paradigm of stochastic resonance
can quite generally and roughly be seen as the optimal amplification of a weak pe-
riodic signal in a dynamical system triggered by random forcing. In this section, we
shall briefly deviate from the presentation of our mathematical approach of optimal
tuning by large deviations methods, illustrate the ubiquity of the phenomenon of](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-42-320.jpg)
![22 1. HEURISTICS
stochastic resonance. We will briefly discuss some prominent examples of dynami-
cal systems arising in different areas of natural sciences in which it occurs, following
several big reviews on stochastic resonance from the point of view of natural sciences
such as [1, 43, 44, 79, 108]. We refer the reader to these references for ample fur-
ther information on a huge number of examples where stochastic resonance appears.
Finally we will briefly comment on computational aspects of stochastic resonance
that are important in particular in high dimensional applications.
1.5.1. Resonant activation and Brownian ratchets. The two popular
examples we mention here are elementary realizations of transition phenomena
corresponding roughly to our paradigm of an overdamped Brownian particle in a
potential landscape subject to weak periodic variation of some parameters. Here
we face the examples of one-well potentials resp. asymmetric periodic multi-well
potentials.
The effect of the so-called resonant activation arises in the simple situation
in which an overdamped Brownian particle exits from a single potential well with
randomly fluctuating potential barrier. In the case we consider the potential barrier
can be considered to undergo weak periodic deterministic fluctuations in contrast.
Even in the simplest situation, in which the height of the potential barrier is given
by a Markov chain switching between two states, one can observe a non-linear
dependence of the mean first exit time from the potential well and the intensity of
the switching (see e.g. Doering and Elston [28]).
Noise induced transport in Brownian ratchets addresses the directed motion of
the Brownian particle in a spatially asymmetric periodic potential having the shape
of a long chain of downward directed sawtooths of equal length. It arises as another
exit time phenomenon, since random exits over the lower potential barrier on the
right hand side of the particle’s actual position are highly favored. For instance in
the context of an electric conductor, this effect creates a current in the downward
direction indicated, see Doering et al. [28, 27] and Reimann [91]. An important
application of this effect is the biomolecular cargo transport, see e.g. Elston and
Peskin [35] and Vanden–Eijnden et al. [80, 26].
1.5.2. Threshold models and the Schmitt trigger. Models of stochastic
resonance based on a bistable weakly periodic dynamical system of the type (0.1)
are often referred to as dynamical models in contrast to the so-called non-dynamical
or threshold models. These are models usually consisting of a biased deterministic
input which may be periodic or not, and a multi-state output. In the simplest
situation, the output takes a certain value as the input crosses a critical threshold.
The simplest model of this type is the Schmitt trigger, an electronic device studied
first by Fauve and Heslot [38] and Melnikov [76] (see also [1, 43, 69, 70, 74]).
It is given by a well-known electronic circuit, characterized by a two-state output
and a hysteretic loop. The circuit is supplied with the input voltage w = wt, which
is an arbitrary function of time. In the ideal Schmitt trigger the output voltage
Y = Yt has only two possible values, say −V and V . Let w increase from −∞.
Then Y = V until w reaches the critical voltage level V+. As this happens, the
output jumps instantaneously to the level −V . Decreasing w does not affect the
output Y until w reaches the critical voltage V−. Then Y jumps back. Therefore,
the Schmitt trigger is a bistable system with hysteresis, see Figure 1.14. The width
of the hysteresis loop is V+ − V−. Applying a periodic voltage of small amplitude a](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-43-320.jpg)
![1.5. STOCHASTIC RESONANCE IN MODELS FROM ELECTRONICS TO BIOLOGY 23
Figure 1.14. The input-output characteristic (hysteresis loop) of
the Schmitt trigger.
and period T 0, for example, to V+, we periodically modulate the critical level.
After adding a random noise at the input, the system is able to jump between the
two states ±V . As in the example of glacial cycles we can consider a discontinuous
modulation, for instance given by V+(t) = a sign(sin (2πt
T )). The whole picture is
now similar to the one in (1.3). Here the periodic modulation of the reference
voltage corresponds to the tilting of the potential wells.
Fauve and Heslot [38] studied the power spectrum of the system and, as in the
glacial cycle example, established that the energy carried by the spectral component
of Y at a given driving frequency has a local maximum for a certain intensity of
the input noise.
The Schmitt trigger provides another interpretation to the phenomenon of sto-
chastic resonance. A system displaying stochastic resonance can be considered as
a random amplifier. The weak periodic signal which cannot be detected in the
absence of noise, can be successfully recovered if the system (the Schmitt trigger
or (1.3)) is appropriately tuned. In other words, the weak underlying periodicity is
exhibited at appropriately chosen non-zero levels of noise, and gets lost if noise is
either too small or too large.
To date, the most important application area of threshold models is neural
dynamics (see Bulsara et al. [17], Douglass et al. [29], Patel and Kosko [85]) and
transmission of information (see Neiman et al. [81], Simonotto et al. [99], Stocks
[101], Moss et al. [79]). The recent book [73] by McDonnell et al. gives a very
complete account on the theory of non-dynamic or threshold stochastic resonance.
1.5.3. The paddlefish. In this well known and frequently discussed example
stochastic resonance appears in the noise-enhanced feeding behavior of the pad-
dlefish Polyodon spathula (see Greenwood et al. [47], Russel et al. [95], Freund et
al. [42]). This species of fish lives in the Midwest of the United States and in the
Yangtze River in China, and feeds on the zooplankton Daphnia. To detect its prey
animals under limited visibility conditions at river bottoms, the paddlefish uses the
long rostrum in front of its mouth as an electrosensory antenna. The frequency
range of sensitivity of the rostrum’s electroreceptors well overlaps with the range
of frequencies produced by the prey. Roughly, the capture probability is observed
as a function of the position of the prey relative to the rostrum. In experiments,
external noise was generated by electrodes connected to an electric noise genera-
tor. It was observed that the spatial distribution and number of strike locations](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-44-320.jpg)
![24 1. HEURISTICS
is a function of the external noise intensity, with a maximum of captures of more
distant plankton at some optimal external noise intensity. If experimentally noisy
electric signals improve the sensitivity of the electroreceptors, nature itself should
also provide sources of noise. In [95] it was conjectured that, besides the signal,
such a noise might be produced by the populations of prey animals themselves.
In [42] this conjecture was confirmed by measurements of the noise strength pro-
duced by single Daphnia in the vicinity of a swarm. In the simple quantitative
approaches, quality of tuning is measured by Fisher information, a concept that
may be comparable to the entropy notions in Chapter 3, Section 3.2.
1.5.4. The FitzHugh–Nagumo system. A more detailed modeling of neu-
ral activities of living systems underlies this well known and studied example. It
deals with action potentials and electric currents transmitted through systems of
ion channels provided by the axons in neural networks, triggered by their mutual
interaction and the interaction of the system with the biological environment. Neu-
rons communicate with each other or with muscle cells by means of electric signals.
Each single neuron can be modeled as an excitable dynamical system: in the rest
state characterized by a negative potential gap with respect to the extracellular
environment, no current flows through the membrane of the neuronal cell. If this
threshold potential barrier disappears due to noisy perturbations created by the
environment (neighboring cells, external field), ion channels through the membrane
are opened and currents appear in form of a spike or firing, followed by a deter-
ministic recovery to the rest state. During a finite (refractory) time interval, the
membrane potential is hyperpolarized by the current flow, and any firing impossi-
ble. The theory that captures the above-mentioned features of neuronal dynamics,
including the finite refractory time, is described by the FitzHugh–Nagumo (FHN)
equations (see Kanamaru et al. [63]). In the diffusively coupled form, a system of
N coupled neurons is described by the system of equations (see [63])
τu̇i(t) =
− vi +
ui −
u3
i
3
+ S(t)
+
√
ε Ẇi(t) +
1
N
N
j=1
(ui − uj),
v̇i(t) = ui − βvi + γ.
Here ui describes the membrane potential of neuron i, vi a variable describing
whether and to which degree neuron i is in the refractory interval of time after
firing, S describes an external periodic pulse acting on the potential levels, while
W1, . . . , Wn is a vector of independent Brownian motions. Finally, τ, β, ε and γ
are system parameters. In the infinite particle limit, the system becomes a sto-
chastic partial differential equation. Roughly, total throughput current will be a
function of the model parameters, and stochastic resonance appears as its optimal
value for a suitable parameter choice (synchronization). The paper by Wiesenfeld
et al. [109] reports about a much simplified form of this system, in which action
potentials of single mechanoreceptor cells of the crayfish Procambarus clarkii are
concerned. The mechanosensory system of the crayfish consists of hairs located
on its tailfan, connected to mechanoreceptor cells. Streaming water moves the hair
and so provides the external excitation that causes the mechanoreceptor cell to fire.
Experimentally (see [109]), a piece of tailfan containing the hair and sensory neuron
was extracted and put into a saline solution environment. Then, periodic pressure](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-45-320.jpg)
![1.5. STOCHASTIC RESONANCE IN MODELS FROM ELECTRONICS TO BIOLOGY 25
modulations and random noise were imposed on the environment. The firings pro-
duced by the mechanoreceptor cell were recorded for different noise levels, and show
clear stochastic resonance peaks as functions of noise intensity. Similar phenomena
are encountered on a much more general basis in the exchange of substances or
information through ionic channels on cell membranes in living organisms.
1.5.5. Physiological systems. The fact that sensory neurons are excitable
systems leading to the FitzHugh–Nagumo equations in the preceding subsection, is
also basic for many suggestions of how to make use of the phenomenon of stochastic
resonance in medicine. Disfunctions arising in sensory organs responsible for hear-
ing, tactile or visual sensations or for balance control could result from relatively
higher sensitivity thresholds compared with those of healthy organs. To raise the
sensitivity level, a natural idea seems to be to apply the right amount of external
noise to these dysfunctional organs, in order to let stochastic resonance effects am-
plify weak signal responses. In experiments reported in Collins et al. [21], local
indentations were applied to the tips of digits of test persons who had to correctly
identify whether a stimulus was presented. Stimuli generated by subthreshold sig-
nals garnished with noise led to improvements in correct identification, with some
optimal noise level indicating a stochastic resonance point. Results like this may be
used for designing practical devices, as for instance gloves, for individuals with ele-
vated cutaneous sensory thresholds. Similarly, randomly vibrating shoe inserts may
help restoring balance control (see Priplata et al. [89]). Stochastic resonance effects
may be used for treating disfunctions of the human blood pressure system (barore-
flex system) featuring a negative feedback between blood pressure and heart rate
resp. width of blood vessels. Blood pressure is monitored by two types of receptors,
for arteries and veins. In Hidaka [54], a weak periodic input was introduced at the
venous blood pressure receptor, whereas noise was added to the arterial receptor.
It was shown that the power of the output signal of the heart rate (measured by
an electrocardiogram) as a function of noise intensity exhibits a bell-shape form,
typical for a curve with a stochastic resonance point. Another group of possible
medical applications of the amplification effects of stochastic resonance is related
to the human brains information processing activity (see Mori and Kai [78]). In
an experiment in Usher and Feingold [103] the effect of stochastic resonance in
the speed of memory retrieval was exhibited. Test persons were proven to perform
single digit calculations (e.g. 7 × 8 =?) significantly faster when exposed to an
optimal level of acoustic noise (via headphones).
1.5.6. Optical systems. In optical systems, stochastic resonance was first
observed in McNamara et al. [75] and Vemuri and Roy [104] in a bidirectional ring
laser, i.e. a ring resonator with a dye as lasing medium. This laser system supports
two meta-stable states realized as modes of the same frequency that travel in oppo-
site directions. They are strongly coupled to each other by the lasing medium, thus
permitting a bistable operation. When the pumping exceeds the lasing threshold,
either clockwise or counterclockwise modes propagate in the laser, with switchings
between those two modes initiated by spontaneous emission in the active medium,
and fluctuations of the pump laser. The net gains of the two propagating modes
in opposite directions can be controlled by an acousto-optical modulator inside the
cavity, which thus can be used both to impose a periodic switching rate between
the modes and to inject noise. Therefore the resulting semiclassical laser equations](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-46-320.jpg)
![26 1. HEURISTICS
are equivalent to those describing overdamped motion of a particle in a periodi-
cally modulated double well potential, as described in the prototypical example of
Section 1.2.
The choice of examples we discussed in more detail is rather selective. The ef-
fects of stochastic resonance have been found in a big number of dynamical systems
in various further areas of the sciences, and studied by a variety of physical mea-
sures of quality of tuning. We just mention a big field of applications in microscopic
systems underlying the laws of quantum mechanics in which intrinsic quantum tun-
neling effects interfere with the interpretation of potential barrier tunneling that
can be seen as causing noise induced transitions in diffusion dynamics. See [108]
for a comprehensive survey. Stochastic resonance has further been observed in
passive optical bistable systems [30], in experiments with magnetoelastic ribbons
[100], in chemical systems [67], as well as in further biological ones [94, 60, 41].
Stochastic resonance may even be observed in more general systems in which the
role of periodic deterministic signals is taken by some other physical mechanisms
(see [108]).
1.5.7. Computational aspects of large deviation theory related to
stochastic resonance. Our theoretical approach aimed at explaining stochastic
resonance conceptually by means of space-time large deviations of weakly periodic
dynamical systems does not touch at all the field of numerical algorithms and sci-
entific computing for stochastic resonance related quantities which become very
important for applications especially in high dimensions. In the framework of the
classical Freidlin–Wentzell theory, first exit time estimates as well as large devia-
tions rates are analytically expressed by the quasi-potential (see Chapter 2) which
can be calculated more or less explicitly for gradient systems. To determine and
minimize the quasi-potential in high-dimensional scenarios is an analytically hardly
accessible task. In Vanden–Eijnden et al. [31, 53] practically relevant algorithms
with numerous applications for this task have been developed. They have been ap-
plied to various problems in different areas of application of stochastic resonance.](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-47-320.jpg)
![CHAPTER 2
Transitions for time homogeneous dynamical
systems with small noise
The trigger of stochastic resonance are diffusion exits from domains, described
by the exit times of randomly perturbed dynamical systems from domains of at-
traction of their stable fixed points. These exits can be rigorously understood by
means of large deviations for dynamical systems with small random perturbations,
as has been shown in Freidlin and Wentzell [40]. The core ingredient of our ap-
proach to stochastic resonance are therefore concepts of large deviations. In this
chapter we present a self-contained treatment of the theory of large deviations for
randomly perturbed dynamical systems with additive Gaussian noise. They are
typically given by solutions of stochastic differential equations of the type
(2.1) dXε
t = b(Xε
t ) dt +
√
ε dWt, Xε
0 = x,
with a smooth mapping b of Euclidean space Rd
, and a d-dimensional Brownian
motion W. To fix ideas in the framework of the most important paradigm for our
treatment, think of b = −∇xU for a typically double well potential U on Rd
. By
the action of the Brownian noise, the two well bottoms become meta-stable, and we
are primarily interested in describing the asymptotic exit and transition behavior
of the trajectories of the solution of (2.1) between their domains of attraction,
in the limit ε → 0. Asymptotic exit rates for these random times are derived
via large deviations principles (LDP) for the laws με = P ◦ Xε
of the diffusion
process as ε → 0. In this case, με lives on the space of Rd
-valued continuous
functions defined on R+, endowed with the topology of uniform convergence on
compact subintervals of R+. To derive these LDP, we start with the LDP for the
Wiener process given by Schilder’s theorem. Since by our choice of noise, the
solution of (2.1) depends continuously on Brownian motion, the LDP for W may
be transferred to Xε
via a contraction principle. To establish Schilder’s theorem
from an elementary perspective, we use the Schauder representation of Brownian
motion allowing an elegant approach according to Baldi and Roynette [3]. The
derivation of the asymptotic properties of exit times in the small noise limit follows
the classical treatment by Dembo and Zeitouni [25]. Here the famous Kramers–
Eyring formula for the expected exit time is underpinned with mathematical rigor.
In terms of the pseudopotential V associated with a dissipative drift b it states
that the average time it takes to leave the domain of attraction of one of the
two meta-stable equilibria, say x+, is given to exponential order by exp(V
ε ). Here
V = infy∈χ V (x+, y) is the critical cost to provide for leaving the domain, and χ is
the manifold separating the basins of attraction.
27](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-48-320.jpg)
![28 2. TIME HOMOGENEOUS SYSTEMS
In Section 2.1 we start by representing one-dimensional Brownian motion in its
Schauder decomposition (Theorem 2.5). It allows a direct approach to its regular-
ity properties in terms of Hölder norms on spaces of continuous functions (Theo-
rem 2.6). The key to this elegant and direct approach is Ciesielski’s isomorphism
of normed spaces of continuous functions with sequence spaces via Fourier repre-
sentation (Theorems 2.2 and 2.3). In Section 2.2 we recall general notions and
basic concepts about large deviations, especially addressing their construction from
exponential decay rates of probabilities of basis sets of topologies (Theorem 2.15),
and their transport between different topological spaces via continuous mappings
(contraction principle) (Theorems 2.17, 2.21). In Section 2.3 the elegant Schauder
representation of Brownian motion then allows a derivation of Schilder’s large de-
viation principle (LDP) for Brownian motion (Theorem 2.28) from the elementary
LDP for one-dimensional Gaussian random variables (Theorem 2.22). The Freidlin–
Wentzell theory, extended to locally Lipschitz diffusion coefficients, and culminating
in the LDP for diffusion processes (Theorem 2.36) is presented in Section 2.4. Fi-
nally, in Section 2.5 we follow Dembo and Zeitouni [25] to derive the exit time laws
due to Freidlin and Wentzell [40] for time homogeneous diffusions from domains
of attraction of underlying dynamical systems in the small noise limit (Theorem
2.42).
2.1. Brownian motion via Fourier series
In this section, we shall present Brownian motion in an approach based on
Fourier series with respect to the orthonormal system of Haar functions. This
approach will be seen to open an easy and fast route to large deviations principles
for Brownian motion, the basic noise process added to deterministic dynamical
systems to provide the time homogeneous randomly perturbed dynamical systems
that are the main objects of interest for this section. In fact, we shall present a
direct proof of Schilder’s Theorem which only uses this Fourier series representation
and the large deviation principle for one-dimensional Gaussian variables. The basic
idea of this approach for large deviations on function spaces is triggered by an
observation by Ciesielski according to which smoothness properties of functions in
Hölder spaces can be studied via a universal Banach space isomorphism through
convergence properties of sequences. We first present Ciesielski’s isomorphism.
2.1.1. The Ciesielski isomorphism of Hölder and sequence spaces.
For 0 α ≤ 1, let Cα
([0, 1], R) be the space of all α-Hölder continuous functions
f : [0, 1] → R starting at zero, f(0) = 0. This space is a Banach space endowed
with the Hölder norm
fα = sup
0≤ts≤1
|f(t) − f(s)|
|t − s|α
.
Denote, moreover, by l∞
(N0, R) the Banach space of all bounded real valued se-
quences η = (ηn)n≥0 endowed with the norm η∞ = supn≥0 |ηn|. We call two
Banach spaces isomorphic if there exists a one-to-one linear map between the
spaces. By means of Fourier decomposition of Hölder continuous functions with
the Schauder basis we first prove that Cα
([0, 1], R) and l∞
(N0, R) are isomorphic.](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-49-320.jpg)
![2.1. BROWNIAN MOTION VIA FOURIER SERIES 29
For this purpose we introduce the Haar functions. For t ∈ [0, 1] let χ0(t) ≡ 1 and
χ2k+l(t) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
√
2k if t ∈ 2l
2k+1 , 2l+1
2k+1
,
−
√
2k if t ∈ 2l+1
2k+1 , 2l+2
2k+1
, k ≥ 0, l = 0, 1, . . . , 2k
− 1,
0 otherwise.
The family (χn)n≥0 is a complete orthonormal system of L2
([0, T], Rd
) . The
Schauder functions are just the primitives of the Haar system, given for t ∈ [0, 1]
by
φn(t) =
t
0
χn(s) ds.
If f ∈ Cα
([0, 1], R) possesses a square integrable density ˙
f ∈ L2
([0, 1], R) so that for
t ∈ [0, 1] we have
f(t) =
t
0
˙
f(s) ds,
we can write
˙
f =
∞
n=0
χn, ˙
fχn,
and therefore
f =
∞
n=0
χn, ˙
fφn.
Indeed, due to the fact that for k ≥ 0 fixed, and 0 ≤ l1, l2 ≤ 2k
− 1, l1 = l2,
the supports of the functions φ2k+l1
and φ2k+l2
are disjoint and the functions are
uniformly bounded by 2− k
2 −1
, we may estimate for K ∈ N, q ≥ p ≥ 2K
by means
of Cauchy–Schwarz’ inequality
(2.2)
q
n=p
χn, ˙
fφn
≤
∞
k=K
2k
−1
l=0
χ2k+l, ˙
fφ2k+l
≤
∞
k=K
sup
0≤l≤2k−1
|χ2k+l, ˙
f| · 2− k
2 −1
≤
∞
k=K
1
0
˙
f2
(s) ds
1
2
· 2− k
2 −1
.
This clearly implies the convergence of the series in the uniform norm. We shall
now see by following Ciesielski [20] that this representation may be extended to
Hölder spaces. For this purpose denote for n = 2k
+ l
χn, ˙
f =
√
2k 2f
2l + 1
2k+1
− f
2l + 2
2k+1
− f
2l
2k+1
.
This just gives the integral of the function χn with respect to f as an integrator.
Lemma 2.1. Let 0 α ≤ 1, and let f ∈ Cα
([0, 1], R). Then
f =
∞
n=0
χn, ˙
fφn,
with convergence with respect to the uniform norm.](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-50-320.jpg)
![30 2. TIME HOMOGENEOUS SYSTEMS
Proof. It can be seen easily that f may be approximated in the uniform norm
by a sequence (fm)m∈N of functions possessing square integrable densities ( ˙
fm)m∈N,
and with α-Hölder norms bounded by the one of f. Take for instance a sequence
obtained from f by smoothing with a sequence of smooth approximations of the
unit. More precisely, let φ: [−1, 1] → R+ be a C∞
function such that
1
−1
φ(x) dx =
1. For m ∈ N, let φm(·) = mφ(m ·), and fm(t) =
1
−1
f(t − x)φm(x) dx, t ∈ [0, 1]
(here we assume f to be trivially extended continuously to [−1, 2] by constant
branches). Obviously, (fm)m∈N converges to f in the uniform norm, and for each
m, fm possesses a square integrable density ˙
fm. Moreover, for s, t ∈ [0, 1], m ∈ N,
α ∈ (0, 1],
|fm(t) − fm(s)|
|t − s|α
≤
1
−1
|f(t − x) − f(s − x)|
|t − s|α
φm(x) dx,
hence fmα ≤ fα.
Since we know from the above discussion that the desired representations hold
for fm for all m ∈ N, a dominated convergence argument shows that we have to
prove
(2.3) sup
m∈N
q
n=p
χn, ˙
fmφn
→ 0 as q ≥ p → ∞.
To do this, we have to modify the estimate (2.2) a bit. In fact, for any m ∈ N, and
K ∈ N, q ≥ p ≥ 2K
we have
(2.4)
q
n=p
χn, ˙
fmφn
≤
∞
k=K
2k
−1
l=0
χ2k+l, ˙
fmφ2k+l
≤
∞
k=K
sup
0≤l≤2k−1
|χ2k+l, ˙
fm|2− k
2 −1
≤
∞
k=K
fmα2−αk
≤ fα ·
∞
k=K
2−αk
.
The latter expression obviously converges to 0 as K → ∞. This completes the
proof.
The following Theorem states that Cα
([0, 1], R) and l∞
(N0, R) are isomorphic.
Theorem 2.2. Let 0 α 1. For n ∈ N0 let
cn(α) =
1, n = 0,
2k(α− 1
2 )+α−1
, n = 2k
+ l, k ≥ 0, 0 ≤ l ≤ 2k
− 1.
Define
Tα : Cα
([0, 1], R) → l∞
(N0, R), f → (cn(α) χn, ˙
f)n≥0.
Then
T−1
α : l∞
(N0, R) → Cα
([0, 1], R), (ηn)n≥0 →
∞
n=0
ηn
cn(α)
φn,](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-51-320.jpg)
![32 2. TIME HOMOGENEOUS SYSTEMS
The spaces we are ultimately interested in are those in which almost all sample
paths of the Brownian motion are living. We therefore have to extend the isomor-
phism of Theorem 2.2 to the following subspaces of Hölder continuous functions.
For 0 α ≤ 1, let Cα
0 ([0, 1], R) be the subspace of Cα
([0, 1], R) composed of all
functions f for which f(0) = 0 and
lim
δ→0
sup
0≤st≤1,
|t−s|≤δ
|f(t) − f(s)|
|t − s|α
= 0.
The isomorphism of Theorem 2.2 will then be restricted to the subspace c0(N0, R)
of all sequences η = (ηn)n≥0 in l∞
(N0, R) which converge to 0 as n → ∞ as a target
space. The following Theorem holds.
Theorem 2.3. Let 0 α 1. Let (cn(α))n≥0 be defined as in Theorem 2.2.
Define
Tα,0 : Cα
0 ([0, 1], R) → c0(N0, R), f → (cn(α) χn, ˙
f)n≥0.
Then
T−1
α,0 : c0(N0, R) → Cα
0 ([0, 1], R), (ηn)n≥0 →
∞
n=0
ηn
cn(α)
φn.
The mapping Tα,0 is an isomorphism, and for the operator norms we have the
following inequalities
Tα,0 = 1, T−1
α,0 ≤
2
(2α − 1)(21−α − 1)
.
Proof. Note first that (2.5) can be strengthened by definition to read
(2.9) |χn, ˙
f| ≤
1
cn(α)
sup
0≤st≤1,
|t−s|≤2−k−1
|f(t) − f(s)|
|t − s|α
.
Hence, by definition of Cα
0 ([0, 1], R), we obviously obtain that Tα,0 is well defined.
To prove that also T−1
α,0 is well defined, we just have a closer inspection of the
arguments that led to the operator norm inequality in the proof of Theorem 2.2.
First, note that for η = (ηn)n≥0 ∈ c0(N0, R), with f = T−1
α,0(η)
(2.10) |f(t)−f(s)| ≤ |η0(α)|·|t−s|+
∞
k=0
2k
−1
l=0
1
c2k+l(α)
|η2k+l|·|φ2k+l(t)−φ2k+l(s)|.
Now choose again k0 ≥ 0 such that
2−k0−1
|t − s| ≤ 2−k0
,
and denote τn = supk≥n |ηk|. Then for 0 ≤ k k0
(2.11)
2k
−1
l=0
1
c2k+l(α)
|η2k+l| · |φ2k+l(t) − φ2k+l(s)|
= sup
0≤l≤2k−1
1
c2k+l(α)
|η2k+l| · |φ2k+l(t) − φ2k+l(s)|
≤ τ2k · 2−k(α− 1
2 )−α+1
· 2
k
2 |t − s|
≤ τ2k · 2k(1−α)−α+1−k0(1−α)
· |t − s|α
= τ2k · (21−α
)(1+k−k0)
· |t − s|α
,](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-53-320.jpg)
![2.1. BROWNIAN MOTION VIA FOURIER SERIES 33
while for k ≥ k0
(2.12)
2k
−1
l=0
1
c2k+l(α)
|η2k+l| · |φ2k+l(t) − φ2k+l(s)|
≤ τ2k0 · 2−k(α− 1
2 )−α+1
· 2− k
2
≤ τ2k0 · 2−kα−α+1+(k0+1)α
· |t − s|α
= τ2k0 · (2α
)(k0−k)
· |t − s|α
.
Hence (2.10), (2.11) and (2.12) imply
|f(t) − f(s)|
|t − s|α
≤ η∞ · |t − s|1−α
+
k≤k0
τ2k · (21−α
)(1+k−k0)
+
1
2α − 1
τ2k0 .
Now k0 → ∞ as |t − s| → 0. This, however, entails that f ∈ Cα
0 ([0, 1], R). All
arguments used in the proof of Theorem 2.2 to show the inequalities for the operator
norms are valid here. Just note that the function f(t) = t, 0 ≤ t ≤ 1, belongs to
Cα
0 ([0, 1], R).
2.1.2. The Schauder representation of Brownian motion. We shall now
present an approach to the study of one-dimensional Brownian motion which is close
to Wiener’s representation of Brownian motion by Fourier series with trigonometric
functions as a basis. Our basis will be given by the Haar functions and their
primitives. In fact, the trajectories of Brownian motion will be described just
as in the preceding section continuous functions were isomorphically described by
sequences. Given a Brownian motion W indexed by the unit interval, with the
same notation as in the preceding section we write it sample by sample as a series
with coefficients χn, Ẇ, n ∈ N0. Due to the scaling properties and the structure
of Haar functions, these random coefficients are i.i.d. standard normal random
variables. This, in turn, allows us to construct Brownian motion indexed by the
unit interval by taking any sequence of i.i.d. standard normal variables (Zn)n∈N0
on a probability space (Ω, F, P), and defining the stochastic process
(2.13) Wt =
∞
n=0
Znφn(t), t ∈ [0, 1].
To get information about the quality of convergence of this Fourier series, we need
to control the size of the random sequence (Zn)n∈N0
in the following Lemma.
Lemma 2.4. Let (Zn)n≥2 be a sequence of standard gaussian random variables.
There exists a real valued random variable C such that the inequality
(2.14) sup
n≥2
|Zn(ω)| ≤ C(ω)
√
ln n
holds true P-a.s.
Proof. For x ≥ 1 and n ≥ 2 we have
P(|Zn| ≥ x) =
2
π
∞
x
e− u2
2 du ≤
2
π
∞
x
ue− u2
2 du =
2
π
e− x2
2 .
Hence for any β 1
P
|Zn| ≥
2β ln n
≤
2
π
e−β ln n
=
2
π
1
nβ
.](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-54-320.jpg)
![gallows and the cannon will be useless. This is no longer a mere
Utopia, but a new and definite system of life, toward which mankind
is progressing with ever increasing rapidity.
But when will it come?
Eighteen hundred years ago Christ, in answer to this question,
replied that the end of the present world—that is, of the pagan
system—would come when the miseries of man had increased to
their utmost limit; and when, at the same time, the good news of
the Kingdom of Heaven—that is, of the possibility of a new system,
one not founded upon violence—should be proclaimed throughout
the earth.[20]
But of that day and hour knoweth no man, no, not the angels of
heaven, but my Father only,[21]
said Christ. Watch therefore: for ye
know not what hour your Lord doth come.
When will the hour arrive? Christ said that we cannot know. And for
that very reason we should hold ourselves in readiness to meet it, as
the goodman should watch his house against thieves, or like the
virgins who await with their lamps the coming of the bridegroom;
and, moreover, we should work with all our might to hasten the
coming of that hour, as the servants should use the talents they
have received that they may increase.[22]
And there can be no other answer. The day and the hour of the
advent of the Kingdom of God men cannot know, since the coming
of that hour depends only on men themselves.
The reply is like that of the wise man who, when the traveler asked
him how far he was from the city, answered, Go on!
How can we know if it is still far to the goal toward which humanity
is aiming, when we do not know how it will move toward it; that it
depends on humanity whether it moves steadily onward or pauses,
whether it accelerates or retards its pace.
All that we can know is what we who form humanity should or
should not do in order to bring about this Kingdom of God. And that](https://image.slidesharecdn.com/2626113-250609220654-c96984ba/85/Stochastic-Resonance-A-Mathematical-Approach-In-The-Small-Noise-Limit-Samuel-Herrmann-83-320.jpg)
Stochastic Resonance A Mathematical Approach In The Small Noise Limit Samuel Herrmann
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- 5. Mathematical Surveys and Monographs Volume 194 American Mathematical Society Stochastic Resonance A Mathematical Approach in the Small Noise Limit Samuel Herrmann Peter Imkeller Ilya Pavlyukevich Dierk Peithmann
- 6. Stochastic Resonance A Mathematical Approach in the Small Noise Limit
- 8. Mathematical Surveys and Monographs Volume 194 Stochastic Resonance A Mathematical Approach in the Small Noise Limit Samuel Herrmann Peter Imkeller Ilya Pavlyukevich Dierk Peithmann American Mathematical Society Providence, Rhode Island
- 9. EDITORIAL COMMITTEE Ralph L. Cohen, Chair Robert Guralnick Michael A. Singer Benjamin Sudakov Michael I. Weinstein 2010 Mathematics Subject Classification. Primary 60H10, 60J60; Secondary 34D45, 37H10, 60F10, 60J70, 60K35, 86A10. For additional information and updates on this book, visit www.ams.org/bookpages/surv-194 Library of Congress Cataloging-in-Publication Data Herrmann, Samuel, author. Stochastic resonance : a mathematical approach in the small noise limit / Samuel Herrmann, Peter Imkeller, Ilya Pavlyukevich, Dierk Peithmann. pages cm. — (Mathematical surveys and monographs ; volume 194) Includes bibliographical references and index. ISBN 978-1-4704-1049-0 (alk. paper) 1. Stochastic partial differential equations. 2. Diffusion processes. 3. Stability. I. Imkeller, Peter, 1951– author. II. Pavlyukevich, Ilya, 1974– author. III. Peithmann, Dierk, 1972– author. IV. Title. QA274.25.H47 2014 519.23—dc23 2013034700 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to reprint-permission@ams.org. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 19 18 17 16 15 14
- 10. Contents Preface vii Introduction ix Chapter 1. Heuristics of noise induced transitions 1 1.1. Energy balance models of climate dynamics 1 1.2. Heuristics of our mathematical approach 6 1.3. Markov chains for the effective dynamics and the physical paradigm of spectral power amplification 14 1.4. Diffusions with continuously varying potentials 18 1.5. Stochastic resonance in models from electronics to biology 21 Chapter 2. Transitions for time homogeneous dynamical systems with small noise 27 2.1. Brownian motion via Fourier series 28 2.2. The large deviation principle 37 2.3. Large deviations for Brownian motion 44 2.4. The Freidlin–Wentzell theory 50 2.5. Diffusion exit from a domain 59 Chapter 3. Semiclassical theory of stochastic resonance in dimension 1 69 3.1. Freidlin’s quasi-deterministic motion 69 3.2. The reduced dynamics: stochastic resonance in two-state Markov chains 78 3.3. Spectral analysis of the infinitesimal generator of small noise diffusion 91 3.4. Semiclassical approach to stochastic resonance 114 Chapter 4. Large deviations and transitions between meta-stable states of dynamical systems with small noise and weak inhomogeneity 133 4.1. Large deviations for diffusions with weakly inhomogeneous coefficients134 4.2. A new measure of periodic tuning induced by Markov chains 144 4.3. Exit and entrance times of domains of attraction 154 4.4. The full dynamics: stochastic resonance in diffusions 169 Appendix A. Supplementary tools 177 Appendix B. Laplace’s method 179 Bibliography 183 Index 189 v
- 12. Preface Stochastic resonance is a phenomenon arising in many systems in the sciences in a wide spectrum extending from physics through neuroscience to chemistry and biology. It has attracted an overwhelming attendance in the science literature for the last two decades, more recently also in the mathematics literature. It is generally understood as the optimal amplification of a weak periodic signal in a dynamical system by random noise. This book presents a mathematical approach of stochastic resonance in a well defined framework. We consider weakly periodic systems in arbitrary finite di- mension with additive noise of small amplitude ε. They possess two domains of attraction of stable equilibria separated by a manifold marking a barrier. Both the geometry of the attraction domains as well as the barrier height are not scaled with the amplitude parameter ε. Therefore, in contrast to other approaches, noise induced random transitions in our model happen on time scales given by the expo- nential of the quotient of barrier height and noise amplitude (Kramers’ times), and are due to large deviations. Our analysis is therefore based on a new space-time large deviations principle for the system’s exit and transition dynamics between different domains of attraction in the limit of small ε. It aims at the description of an optimal interplay between large period length T of the weak periodic motion and noise amplitude ε. Optimization is done with respect to appropriate measures of quality of tuning of the stochastic system to the periodic input. The two principal messages of the book are these. First we show that—already in space dimension one—the classical physical measures of quality of periodic tun- ing such as the spectral power amplification or signal-to-noise ratio, due to the impact of small random oscillations near the equilibria, are not robust with respect to dimension reduction. Comparing optimal tuning rates for the unreduced (dif- fusion) model and the associated reduced (finite state Markov chain) model one gets essentially different tuning scenarios. We therefore propose—in arbitrary fi- nite space dimension—measures of quality of periodic tuning based uniquely on the transition dynamics and show that these measures are robust. Via our central space-time large deviations result they are able to explain stochastic resonance as optimal tuning. Concentrating on these more theoretical themes, the book sheds some light on the mathematical shortcomings and strengths of different concepts used in theory and application of stochastic resonance. It does not aim at a comprehensive pre- sentation of the many facets of stochastic resonance in various areas of sciences. In particular it does not touch computational aspects relevant in particular in high dimensions where analytical methods alone are too complex to be of practical use any more. vii
- 13. viii PREFACE With this scope the book addresses researchers and graduate students in math- ematics and the sciences interested in stochastic dynamics, in a quite broad sense, and wishing to understand the conceptual background of stochastic resonance, on the basis of large deviations theory for weakly periodic dynamical systems with small noise. Chapter 1 explains our approach on a heuristic basis on the background of paradigmatic examples from climate dynamics. It is accessible to a readership without a particular mathematical training. Chapter 2 provides a self-contained treatment of the classical Freidlin–Wentzell theory of diffusion exit from domains of attraction of dynamical systems in the simpler additive noise setting starting from a wavelet expansion of Brownian motion. It should be accessible to readers with basic knowledge of stochastic processes. In Chapter 3 based on an approach from the perspective of semi-classical analysis, i.e. spectral theory of infinitesimal gen- erators of diffusion processes, the conceptual shortcomings of the classical physical concepts of stochastic resonance are presented. In Chapter 4 the Freidlin–Wentzell theory is extended to the non-trivial setting of weakly time-periodic dynamical sys- tems with noise, and concepts of optimal tuning discussed which avoid the defects of the classical notions. Both Chapters are accessible on the basis of the background knowledge presented in Chapter 2.
- 14. Introduction Speaking about noise we usually mean something that deteriorates the opera- tion of a system. It is understood as a disturbance, a random and persistent one, that obscures or reduces the clarity of a signal. In nonlinear dynamical systems, however, noise may play a very constructive role. It may enhance a system’s sensitivity to a small periodic deterministic signal by amplifying it. The optimal amplification of small periodic signals by noise gives rise to the ubiquitous phenomenon of stochastic resonance (SR) well studied in a plethora of papers in particular in the physical and biological sciences. This book presents a mathematical approach to stochastic resonance in a well defined particular mathematical framework. We consider weakly periodic systems with additive noise of small amplitude ε. The systems possess two domains of attraction of stable equilibria separated by a manifold marking a barrier. Both the geometry of the attraction domains as well as the barrier height are not subject to scalings with the amplitude parameter ε. Therefore, as opposed to other approaches, noise induced random transitions in our model happen on time scales of Kramers’ law, i.e. they are exponential in the quotient of barrier height and noise amplitude, and are due to large deviations. Our analysis is therefore based on a new large deviations principle of the systems’ exit and transition dynamics between different domains of attraction in the limit of small ε. It aims at the description of an optimal interplay between large period length T of the weak periodic motion and noise amplitude ε, where optimization is done with respect to appropriate measures of quality of response of the stochastic system to the periodic input. We will be uniquely concerned with the well founded and self contained presentation of this mathematical approach mainly based on a space-time extension of Freidlin– Wentzell’s theory of large deviations of noisy dynamical systems, first on a heuristic and then on a mathematically rigorous level. The two principal messages of the book are these. First we show that — already in space dimension one — the classical physical measures of quality of periodic tuning such as the spectral power amplification, due to the phenomenon of the small oscillations catastrophe, are not robust with respect to dimension reduction. Comparing optimal tuning rates for the diffusion processes and the finite state Markov chains retaining the models’ essentials one gets essentially different results (Chapter 3, Theorems 3.50, 3.53). We therefore propose — in arbitrary finite space dimension — measures of quality of periodic tuning based uniquely on the transition dynamics and show that these measures are robust and, via a crucial large deviations result, are able to explain stochastic resonance as optimal tuning (Chapter 4, Theorems 4.19, 4.29, 4.31). Concentrating on these more theoretical themes, the book sheds some light on the mathematical shortcomings and strengths of different concepts used in theory and application of stochastic resonance, in a well defined framework. It does not aim at ix
- 15. x INTRODUCTION a comprehensive presentation of the many facets of stochastic resonance in various areas of sciences (a sample will be briefly discussed in Chapter 1, Section 1.5). In particular it does not touch computational aspects relevant in particular in high dimensions where analytical methods alone are too complex to be of practical use any more (for an incomplete overview of stochastic resonance from a computational dynamics perspective see also Chapter 1, Section 1.5). We now explain briefly our motivation and approach. The most prominent and one of the first examples in which phenomena related to stochastic resonance were observed is given by energy balance models of low dimensional conceptual climate dynamics. It was employed for a qualitative explanation of glacial cycles in earth’s history, i.e. the succession of ice and warm ages observed in paleoclimatic data, by means of stochastic transitions between cold and warm meta-stable climates in a dynamical model. It will be discussed in more detail in Chapter 1. The model proposed by Nicolis [83] and Benzi et al. [6] is based on the balance between aver- aged absorbed and emitted radiative energy and leads to a deterministic differential equation for averaged global temperature T of the form Ṫ(t) = b(t, T(t)). The explicit time dependence of b captures the influence of the solar constant that undergoes periodic fluctuations of a very small amplitude at a very low frequency. The fluctuations are due to periodic changes of the earth’s orbital parameters (Mi- lankovich cycles), for instance a small variation of the axial tilt that arises at a frequency of roughly 4 × 10−4 times per year, and coincide roughly with the ob- served frequencies of cold and warm periods. For frozen t the nonlinear function b(t, T) describes the difference between absorbed radiative energy as a piecewise linear function of the temperature dependent albedo function a(T) and emitted ra- diative energy proportional to T4 due to the Stefan–Boltzmann law of black body radiators. In the balance for relevant values of T it can be considered as negative gradient (force) of a double well potential, for which the two well bottoms corre- spond to stable temperature states of glacial and warm periods. The evolution of temperature in the resulting deterministic dynamical system is analogous to the motion of an overdamped physical particle subject to the weakly periodic force field of the potential. Trajectories of the deterministic system relax to the stable states of the domain of attraction in which they are started. Only the addition of a stochastic forcing to the system allows for spontaneous transitions between the different stable states which thus become meta-stable. In a more general setting, we study a dynamical system in d-dimensional Eu- clidean space perturbed by a d-dimensional Brownian motion W, i.e. we consider the solution of the stochastic differential equation (0.1) dXε t = b t T , Xε t dt + √ ε dWt, t ≥ 0. One of the system’s important features is that its time inhomogeneity is weak in the sense that the drift depends on time only through a re-scaling by the time parameter T = T(ε) which will be assumed to be exponentially large in ε. This corresponds to the situation in Herrmann and Imkeller [50] and is motivated by the well known Kramers–Eyring law which was mathematically underpinned by the Freidlin–Wentzell theory of large deviations [40]. The law roughly states that the expected time it takes for a homogeneous diffusion to leave a local attractor e.g.
- 16. INTRODUCTION xi across a potential wall of height v 2 is given to exponential order by T(ε) = exp(v ε ). Hence, only in exponentially large scales of the form T(ε) = exp(μ ε ) parametrized by an energy parameter μ we can expect to see effects of transitions between different domains of attraction. We remark at this place that our approach essentially differs from the one by Berglund and Gentz [13]. If b represents a negative potential gradient for instance, their approach would typically not only scale time by T, but also the depths of the potential wells by a function of ε. As a consequence, transitions even for the deterministic dynamical system become possible, and their noise induced transitions happen on time scales of intermediate length. In contrast, in our setting transitions between the domains of attraction of the deterministic system are impossible, and noise induced ones are observed on very large time scales of the order of Kramers’ time, typically as consequences of large deviations. The function b is assumed to be one-periodic w.r.t. time, and so the system described by (0.1) attains period T by re-scaling time in fractions of T. The deterministic system ˙ ξt = b(s, ξt) with frozen time parameter s is supposed to have two domains of attraction that do not depend on s ≥ 0. In the “classical” case of a drift derived from a potential, b(t, x) = −∇xU(t, x) for some potential function U, equation (0.1) is analogous to the overdamped motion of a Brownian particle in a d-dimensional time inhomogeneous double-well potential. In general, trajectories of the solutions of differential equations of this type will exhibit randomly periodic behavior, reacting to the periodic input forcing and eventually amplifying it. The problem of optimal tuning at large periods T consists in finding a noise amplitude ε(T) (the resonance point) which supports this amplification effect in a best possible way. During the last 20 years, various concepts of measuring the quality of periodic tuning to provide a criterion for optimality have been discussed and proposed in many applications from a variety of branches of natural sciences (see Gammaitoni et al. [43] for an overview). Its rigorous mathematical treatment was initiated only relatively late. The first approach towards a mathematically precise understanding of stochas- tic resonance was initiated by Freidlin [39]. To explain stochastic resonance in the case of diffusions in potential landscapes with finitely many minima (in the more general setting of (0.1), the potential is replaced by a quasi-potential related to the action functional of the system), he goes as far as basic large deviations’ theory can take. If noise intensity is ε, in the absence of periodic exterior forcing, the exponen- tial order of times at which successive transitions between meta-stable states occur corresponds to the work to be done against the potential gradient to leave a well (Kramers’ time). In the presence of periodic forcing with period time scale e μ ε , in the limit ε → 0 transitions between the stable states with critical transition energy close to μ will be periodically observed. Transitions with smaller critical energy may happen, but are negligible in the limit. Those with larger critical energy are forbidden. In case the two local minima of the potential have depths V 2 and v 2 , v V , that switch periodically at time 1 2 (in scale T accordingly at time T 2 ), for T larger than e v ε the diffusion will be close to the deterministic periodic function jumping between the locations of the deepest wells. As T exceeds this exponen- tial order, many short excursions to the wrong well during one period may occur. They will not count on the exponential scale, but trajectories will look less and less periodic. It therefore becomes plausible that physicists’ quality measures for periodic tuning which always feature some maximal tuning quality of the random
- 17. xii INTRODUCTION trajectories to the periodic input signal cannot be captured by this phenomenon of quasi-deterministic periodicity at very large time scales. These quality measures, studied in Pavlyukevich [86] and Imkeller and Pavlyuke- vich [59] assess quality of tuning of the stochastic output to the periodic determin- istic input. The concepts are mostly based on comparisons of trajectories of the noisy system and the deterministic periodic curve describing the location of the relevant meta-stable states, averaged with respect to the equilibrium measure (of the diffusion as a space-time process with time component given by uniform motion in the period interval). Again in the simple one-dimensional situation considered above the system switches between a double well potential state U with two wells of depths V 2 and v 2 , v V, during the first half period, and the spatially opposite one U(·) for the second half period. If as always time is re-scaled by T, the total period length is T, and stochastic perturbation comes from the coupling to a white noise of intensity ε. The most important measures of quality studied are the spectral power amplification and the related signal-to-noise ratio, both playing an eminent role in the physical literature (see Gammaitoni et al. [43], Freund et al. [41]). They mainly contain the mean square average in equilibrium of the Fourier component of the solution trajectories corresponding to the input period T, normalized in dif- ferent ways. These measures of quality are functions of ε and T, and the problem of finding the resonance point consists in optimizing them in ε for fixed (large) T. Due to the high complexity of original systems, when calculating the resonance point at optimal noise intensity, physicists usually pass to an effective dynamics description. It is given by a simple caricature of the system reducing the diffu- sion dynamics to the pure inter well motion (see e.g. McNamara and Wiesenfeld [74]). The reduced dynamics is represented by a continuous time two state Markov chain with transition probabilities corresponding to the inverses of the diffusions’ Kramers’ times. One then determines the optimal tuning parameters ε(T) for large T for the approximating Markov chains in equilibrium, a rather simple task. To see that the Markov chain’s behavior approaches the diffusion’s in the small noise limit, spectral theory for the infinitesimal generator is used. The latter is seen to possess a spectral gap between the second and third eigenvalues, and therefore the closeness of equilibrium measures can be well controlled. Surprisingly, due to the importance of small intra well fluctuations, the tuning and resonance pattern of the Markov chain model may differ dramatically from the resonance picture of the diffusion. Subtle dependencies on the geometrical fine structure of the potential function U in the minima beyond the expected curvature properties lead to quite unexpected counterintuitive effects. For example, a subtle drag away from the other well caused by the sign of the third derivative of U in the deep well suffices to make the spectral power amplification curve strictly increasing in the parameter range where the approximating Markov chain has its resonance point. It was this lack of robustness against model reduction which motivated Her- rmann and Imkeller [50] to look for different measures of quality of periodic tun- ing for diffusion trajectories. These notions are designed to depend only on the rough inter well motion of the diffusion. The measure treated in the setting of one-dimensional diffusion processes subject to periodic forcing of small frequency is related to the transition probability during a fixed time window of exponential length T(ε) = exp(μ ε ) parametrized by a free energy parameter μ according to the
- 18. INTRODUCTION xiii Kramers–Eyring formula. The corresponding exit rate is maximized in μ to ac- count for optimal tuning. The methods of investigation of stochastic resonance in [50] are heavily based on comparison arguments which are not an appropriate tool from dimension 2 on. Time inhomogeneous diffusion processes such as the ones under consideration are compared to piecewise homogeneous diffusions by freezing the potential’s time dependence on small intervals. In Herrmann et al. [51] this approach is extended to the general setting of finite dimensional diffusion processes with two meta-stable states. Since the stochastic resonance criterion considered in [50] is based on transition times between them, our analysis relies on a suitable notion of transition or exit time parametrized again by the free energy parameter μ from T(ε) = exp(μ ε ) as a natural measure of scale. Assume now that the depths of the two equilibria of the potential in analogy to the scenarios considered before are smooth periodic functions of time of period 1 given for one of them by v(t) 2 , and for the other one by the same function with some phase delay (for instance by 1 2 ). Therefore, at time s the system needs energy v(s) to leave the domain of attraction of the equilibrium. Hence an exit from this set should occur at time aμ = inf{t ≥ 0: v(t) ≤ μ} in the diffusion’s natural time scale, in the time re-scaled by T(ε) thus at time aμ · T(ε). To find a quality measure of periodic tuning depending only on the transition dynamics, we look at the probabilities of transitions to the other domain within a time window [(aμ − h)T(ε), (aμ + h)T(ε)] centered at aμ · T(ε) for small h 0. If τ is the random time at which the diffusion roughly reaches the other domain of attraction (to be precise, one has to look at first entrance times of small neighborhoods of the corresponding equilibrium), we use the quantity (again, to be precise, we use the worst case probability for the diffusion starting in a point of a small neighborhood of the equilibrium of the starting domain) Mh (ε, μ) = P τ ∈ [(aμ − h)T(ε), (aμ + h)T(ε)] . To symmetrize this quality measure with respect to switching of the equilibria, we refine it by taking its minimum with the analogous expression for interchanged equilibria. In order to exclude trivial or chaotic transition behavior, the scale parameter μ has to be restricted to an interval IR of reasonable values which we call resonance interval. With this measure of quality, the stochastic resonance point may be determined as follows. We first fix ε and the window width parameter h 0, and maximize Mh (ε, μ) in μ, eventually reached for the time scale μ0(h). Then the eventually existing limit limh→0 μ0(h) will be the resonance point. To calculate μ0(h) for fixed positive h we use large deviations techniques. In fact, our main result consists in an extension of the Freidlin–Wentzell large devia- tions result to weakly time inhomogeneous dynamical systems perturbed by small Gaussian noise which states that lim ε→0 ε ln 1 − Mh (ε, μ) = μ − v(aμ − h), again in a form which is symmetric for switched equilibria. We show that this asymptotic relation holds uniformly w.r.t. μ on compact subsets of IR, a fact which enables us to perform a maximization and find μ0(h). The resulting notion of stochastic resonance is strongly related to the notions of periodic tuning based on interspike intervals (see [49]), which describe the probability distribution for
- 19. xiv INTRODUCTION transitions as functions of time with exponentially decaying spikes near the integer multiples of the forcing periods. It has the big advantage of being robust for model reduction, i.e. the passage from the diffusion to the two state Markov chain describing its reduced dynamics. The techniques needed to prove this main result feature non-trivial extensions and refinements of the fundamental large deviations theory for time homogeneous diffusions by Freidlin–Wentzell [40]. We prove a large deviations principle for the inhomogeneous diffusion (0.1) and further strengthen this result to get uniformity in system parameters. Similarly to the time homogeneous case, where large deviations theory is applied to the problem of diffusion exit culminating in a mathematically rigorous proof of the Kramers–Eyring law, we study the problem of diffusion exit from a domain which is carefully chosen in order to allow for a detailed analysis of transition times. The main idea behind our analysis is that the natural time scale is so large that re-scaling in these units essentially leads to an asymptotic freezing of the time inhomogeneity, which has to be carefully controlled, to hook up to the theory of large deviations of time homogeneous diffusions. The material in the book is organized as follows. In Chapter 1 we give a de- tailed treatment of the heuristics behind our mathematical approach, mostly in space dimension 1. We start by giving a fairly thorough account of the paradigm of glacial cycles which was the historical root of physical models exhibiting sto- chastic resonance. It gives rise to the model equation of a weakly periodically forced dynamical system with noise that can be interpreted as the motion of an overdamped physical particle in a weakly periodically forced potential landscape subject to noise. The heuristics of exit and transition behavior between domains of attraction (potential wells) of such systems based on the classical large deviations theory is explained in two steps: first for time independent potential landscapes, then for potentials switching discontinuously between two anti-symmetric states every half period. Freidlin’s quasi-deterministic motion is seen to not cover the concept of optimal periodic tuning between weak periodic input and randomly am- plified output. They determine stochastic resonance through measures of quality of periodic tuning such as the spectral power amplification or the signal-to-noise ratio. The latter concepts are studied first for finite state Markov chains capturing the dynamics of the underlying diffusions reduced to the meta-stable states, and then for the diffusions with time continuous periodic potential functions. The robustness defect of the classical notions of resonance in passing from Markov chain to diffu- sion is pointed out. Then alternative notions of resonance are proposed which are based purely on the asymptotic behavior of transition times. Finally, examples of systems exhibiting stochastic resonance features from different areas of science are presented and briefly discussed. They document the ubiquity of the phenomenon of stochastic resonance. Our approach is based on concepts of large deviations. Therefore Chapter 2 is devoted to a self-contained treatment of the theory of large deviations for randomly perturbed dynamical systems in finite dimensions. Following a direct and elegant approach of Baldi and Roynette [3], we describe Brownian motion in its Schauder decomposition. It not only allows a direct approach to its regularity properties in terms of Hölder norms on spaces of continuous functions. It also allows a derivation of Schilder’s large deviation principle (LDP) for Brownian motion from the elemen- tary LDP for one-dimensional Gaussian random variables. The key to this elegant
- 20. INTRODUCTION xv and direct approach is Ciesielski’s isomorphism of normed spaces of continuous functions with sequence spaces via Fourier representation. The proof of the LDP for Brownian motion using these arguments is given after recalling general notions and basic concepts about large deviations, especially addressing their construction from exponential decay rates of probabilities of basis sets of topologies, and their transport between different topological spaces via continuous mappings (contrac- tion principle). Since we only consider diffusion processes with additive noise for which Itô’s map is continuous, an appeal to the contraction principle provides the LDP for the homogeneous diffusion processes we study. Finally, we follow Dembo and Zeitouni [25] to derive the exit time laws due to Freidlin and Wentzell [40] for time homogeneous diffusions from domains of attraction of underlying dynamical systems in the small noise limit. Chapter 3 deals with an approach to stochastic resonance for diffusions with weakly time periodic drift and additive noise in the spirit of the associated Mar- kovian semigroups and their spectral theory. This approach, presented in space dimension 1, is clearly motivated by the physical notions of periodic tuning, in particular the spectral power amplification coefficient. It describes the average spectral component of the diffusion trajectories corresponding to the frequency of the periodic input signal given by the drift term. We first give a rigorous account of Freidlin’s quasi-deterministic limiting motion for potential double well diffusions of this type. We then follow the paradigm of the physics literature, in particular NcNamara and Wiesenfeld [74], and introduce the effective dynamics of our weakly periodically forced double well diffusions given by reduced continuous time Markov chains jumping between their two meta-stable equilibria. In this setting, different notions of periodic tuning can easily be investigated. We not only consider the physicists’ favorites, spectral power amplification and signal-to-noise ratio, but also other reasonable concepts in which the energy carried by the Markov chain trajecto- ries or the entropy of their invariant measures are used. Turning to diffusions with weakly time periodic double well potentials and additive noise again, we then de- velop an asymptotic analysis of their spectral power amplification coefficient based on the spectral theory of their infinitesimal generators. It is based on the crucial observation that in the case of double well potentials its spectrum has a gap be- tween the second and third eigenvalue. Therefore we have to give the corresponding eigenvalues and eigenfunctions a more detailed study, in particular with respect to their asymptotic behavior in the small noise limit. Its results then enable us to give a related small noise asymptotic expansion both of the densities of the associated invariant measures as for the spectral power amplification coefficients. We finally compare spectral power amplification coefficients of the Markov chains describing the reduced dynamics and the associated diffusions, to find that in the small noise limit they may be essentially different, caused by the small oscillations catastrophe near the potential wells’ bottoms. This motivates us in Chapter 4 to look for notions of periodic tuning for the so- lution trajectories of diffusions in spaces of arbitrary finite dimension with weakly periodic drifts and additive small noise which do not exhibit this robustness de- fect. We aim at notions related to the maximal probabilities that the random exit or transition times between different domains of attraction of the underlying dy- namical systems happen in time windows parametrized by free energy parameters on an exponential scale. For the two-state Markov chains describing the effective
- 21. xvi INTRODUCTION dynamics of the diffusions with slow and weak time inhomogeneity this optimal transition rate is readily calculated. This concept moreover has the advantage that their related transition times, as well as the corresponding ones for diffusions with a weak noise dependent time inhomogeneity, allow a treatment by methods of large deviations in the small noise limit. We therefore start with a careful extension of large deviations theory to diffusions with slow time inhomogeneity. The central result for the subsequent analysis of their exit times is contained in a large devia- tions principle, uniform with respect to the energy parameter. It allows us in the sequel to derive upper and lower bounds for the asymptotic exponential exit rate from domains of attraction for slowly time dependent diffusions. They combine to the main large deviations result describing the exact asymptotic exponential exit rates for slowly and weakly time inhomogeneous diffusions in the small noise limit. This central result is tailor made for providing the optimal tuning rate re- lated to maximal probability of transition during an exponential time window. We finally compare the resulting stochastic resonance point to the ones obtained for the Markov chains of the reduced dynamics, and conclude that they agree in the small noise limit, thus establishing robustness. In two appendices — for easy reference in the text — we collect some standard results about Gronwall’s lemma and Laplace’s method for integrals with exponential singularities of the integrand.
- 22. CHAPTER 1 Heuristics of noise induced transitions 1.1. Energy balance models of climate dynamics The simple concept of energy balance models stimulated research not only in the area of conceptual climate models, but was at the cradle of a research direc- tion in physics which subsequently took important examples from various domains of biology, chemistry and neurology: it was one of the first examples for which the phenomenon of stochastic resonance was used to explain the transition dynam- ics between different stable states of physical systems. For a good overview see Gammaitoni et al. [43] or Jung [62]. In the end of the 70’s, Nicolis [83] and Benzi et al. [5] almost simultaneously tried stochastic resonance as a rough and qualitative explanation for the glaciation cycles in earth’s history. They were looking for a simple mathematical model appro- priate to explain experimental findings from deep sea core measurements according to which the earth has seen ten glacial periods during the last million years, alter- nating with warm ages rather regularly in periods of about 100 000 years. Mean temperature shifts between warm age and glacial period are reported to be of the order of 10 K, and relaxation times, i.e. transition times between two relatively stable mean temperatures as rather short, of the order of only 100 years. Math- ematically, their explanation was based on an equation stating the global energy balance in terms of the average temperature T(t), where the global average is taken meridionally (i.e. over all latitudes), zonally (i.e. over all longitudes), and annually around time t. The global radiative power change at time t is equated to the differ- ence between incoming solar (short wave) radiative power Rin and outgoing (long wave) radiative power Rout. The power Rin is proportional to the global average of the solar constant Q(t) at time t. To model the periodicity in the glaciation cycles, one assumes that Q undergoes periodic variations due to one of the so-called Milankovich cycles, based on periodic perturbations of the earth’s orbit around the sun. Two of the most prominent cycles are due to a small periodic variation between 22.1 and 24.5 degrees of the angle of inclination (obliquity) of the earth’s rotation axis with respect to its plane of rotation, and a very small periodic change of only about 0.1 percent of the eccentricity, i.e. the deviation from a circular shape, of the earth’s trajectory around the sun. The obliquity cycle has a duration of about 41 000 years, while the eccentricity cycle corresponds to the 100 000 years observed in the temperature proxies from deep sea core measurements mentioned above. They are caused by gravitational influences of other planets of our solar system. In formulas, Q was assumed to be of the form Q(t) = Q0 + b sin ωt, with some constants Q0, b and a frequency ω = 10−5 [1 y ]. 1
- 23. 2 1. HEURISTICS The other component determining the absorbed radiative power Rin is a rough and difficult to model averaged surface albedo of the earth, i.e. the proportion of the solar power absorbed. It is supposed to be just (average) temperature dependent. For temperatures below T, for which the surface water on earth is supposed to have turned into ice, and the surface is thus constantly bright, the albedo is assumed to be constantly equal to a, for temperatures above T, for which all ice has melted, and the surface constantly brown, it is assumed to be given by a constant a a. For temperatures between T and T, the two constant values a and a are simply linearly interpolated in the most basic model. The rough albedo function has therefore the ramp function shape depicted in Figure 1.1. a a 1 0 T T T a(T ) Figure 1.1. The albedo function a = a(T). To have a simple model of Rout, the earth is assumed to behave approximately as a black body radiator, for which the emitted power is described by the Stefan– Boltzmann law. It is proportional to the fourth power of the body’s temperature and is given by γ T4 (t), with a constant γ proportional to the Stefan constant. Hence the simple energy balance equation with periodic input Q on which the model is built is given by (1.1) c d dt T(t) = Q(t) 1 − a(T(t)) − γ T(t)4 , where the constant c describes a global thermal inertia. According to (1.1), (qua- si-) stationary states of average temperature should be given by the solutions of the equation dT (t) dt = 0. If the model is good, they should reasonably well interpret glacial period and warm age temperatures. Graphically, they are given by the intersections of the curves of absorbed and emitted radiative power, see Figures 1.2 and 1.3. As we shall more carefully explain below, the lower (T1(t)) and upper (T3(t)) quasi-equilibria are stable, while the middle one (T2(t)) is unstable. The equilibrium T1(t) should represent an ice age temperature, T3(t) a warm age, while T2(t) is not observed over noticeably long periods. In their dependence on t they should describe small fluctuations due to the variations in the solar constant. But here one encounters a serious problem with this purely deterministic model. If the fluctuation amplitude of Q is small, then we will observe the two disjoint branches of stable solutions T1 and T3 (Figure 1.4).
- 24. 1.1. ENERGY BALANCE MODELS OF CLIMATE DYNAMICS 3 0 T T T Rin Rout power T2(t) T3(t) T1(t) Figure 1.2. Incoming vs. outgoing power. Figure 1.3. Difference of the powers of incoming and outgoing radiation. 0 2 · 105 3 · 105 4 · 105 105 time T3(t) T1(t) Figure 1.4. Equilibrium temperatures T1(t) and T3(t) for small fluctuation amplitude b. However for both branches alone — besides being unrealistically low or high — the difference between minimal and maximal temperature can by no means account for the observed shift of about 10 K, and also the relaxation times are much too long. But the most important shortcoming of the model is the lacking possibility of transitions between the two branches. If we allow the fluctuation amplitude b to be large, the picture is still very unrealistic: There are intervals during which one of the two branches T1 or T3
- 25. 4 1. HEURISTICS 0 2 · 105 3 · 105 4 · 105 105 time T3(t) T1(t) Figure 1.5. Unrealistic equilibrium temperatures T1(t) and T3(t) for large fluctuation amplitude b. vanishes completely, and transitions are still impossible, unless one is willing to accept discontinuous behavior (Figure 1.5). For this reason, Nicolis [83] and Benzi et al. [5] proposed to add a noise term in (1.1). Despite the fact that then negative temperatures become possible, they worked with the equation (1.2) cṪε (t) = Q(t) 1 − a(Tε (t)) − γ Tε (t)4 + √ ε Ẇt, ε 0, where Ẇ is a white noise. In passing to (1.2), stable equilibria of the deterministic system become — approximately at least — meta-stable states of the stochastic system. And more importantly, the unbounded noise process W makes spontaneous transitions (tunneling) between the meta-stable states T1(t) and T3(t) possible. In fact, the random hopping between the meta-stable states immediately exhibits two features which make the model based on (1.2) much more attractive for a qualitative explanation of glaciation cycles: a) the transitions between T1 and T3 allow for far more realistic temperature shifts, b) relaxation times are random, but very short compared to the periods the process solving (1.2) spends in the stable states themselves. But now a new problem arises, which actually provided the name stochastic resonance. If, seen on the scale of the period of Q, ε is too small, the solution may be trapped in one of the states T1 or T3. By the periodic variation of Q, there are well defined periodically returning time intervals during which T1(t) is the more probable state, while T3(t) takes this role for the rest of the time. So if ε is small, the process, initially in T1, may for example fail to leave this state during a whole period while the other one is more probable. The solution trajectory may then look as in Figure 1.6. If, on the other hand, ε is too large, the big random fluctuation may lead to eventual excursions from the actually more probable equilibrium during its domina- tion period to the other one. The trajectory then typically looks like in Figure 1.7. In both cases it will be hard to speak of a random periodic curve. Good tuning with the periodic forcing by Q is destroyed by a random mechanism being too slow or too fast to follow. It turned out in numerous simulations in a number of similar systems that there is, however, an optimal parameter value ε for which the solution
- 26. 1.1. ENERGY BALANCE MODELS OF CLIMATE DYNAMICS 5 0 2 · 105 3 · 105 4 · 105 105 time T ε (t) T3(t) T1(t) Figure 1.6. A typical solution trajectory of equation (1.2) for the small noise amplitude. 0 2 · 105 3 · 105 4 · 105 105 time T ε (t) T3(t) T1(t) Figure 1.7. A typical solution trajectory of equation (1.2) for the large noise amplitude. 0 2 · 10 5 3 · 10 5 4 · 10 5 10 5 time T ε (t) T3(t) T1(t) Figure 1.8. A typical solution trajectory of equation (1.2), the noise amplitude well tuned. curves are well tuned with the periodic input. A typical well tuned curve is shown in Figure 1.8.
- 27. 6 1. HEURISTICS The optimally tuned system is then said to be in stochastic resonance. Nicolis [83] and Benzi et al. [5], by tuning the noise parameter ε to appropriate values, were able to give qualitative explanations for glaciation cycles based on this phenomenon. Stochastic resonance proved to be relevant in other elementary climate models than the energy balance models considered so far. In Penland et al. [87], Wang et al. [107, 106], a two-dimensional stochastic model for a qualitative explanation of the ENSO (El Niño Southern Oscillation) phenomenon also leads to stochastic resonance effects: for certain parameter ranges the model exhibits random tuned transitions between two stable sea surface temperatures. New evidence for the pres- ence of stochastic resonance phenomena in paleo-climatic time series was added by Ganopolski and Rahmstorf [45]. Their paper interprets the GRIP ice core record representing temperature proxies from the Greenland glacier that extend over a period of roughly 90 000 years, and showing the fine structure of the temperature record of the last glacial period. The time series shows about 20 intermediate warm- ings during the last glacial period commonly known under the name of Dansgaard– Oeschger events. These events are clearly marked by rapid spontaneous increases of temperature by about 6K followed by slower coolings to return to the initial basic cold age temperature. It was noted in [45] that a histogram of the number of Dansgaard–Oeschger events with a duration of k · 1480 years, with k = 1, 2, 3, . . . exhibits the typical shape of a stochastic resonance spike train consistent for in- stance with the results of Herrmann and Imkeller [49] for Markov chains describing the effective diffusion dynamics, or Berglund and Gentz [13] for diffusion processes with periodic forcing. 1.2. Heuristics of our mathematical approach The rigorous mathematical elaboration of the concept of stochastic resonance is the main objective of this book. We start its mathematically sound treatment by giving a heuristical outline of the main stream of ideas and arguments based on the methods of large deviations for random dynamical systems in the framework of the Freidlin–Wentzell theory. Freidlin [39] is able to formulate Kramers’ [65] very old seminal approach mathematically rigorously in a very general setting, and this way provides a lower estimate for the good tuning (see also the numerical results by Milstein and Tretyakov [77]). To obtain an upper estimate, we finally argue by embedding time discrete Markov chains into the diffusion processes that describe the effective dynamics of noise induced transitions. Optimal tuning results obtained for the Markov chains will then be transferred to the original diffusion processes. To describe the idea of our approach, let us briefly return to our favorite ex- ample explained in the preceding section. Recall that the function f(t, T) = Q(t) 1 − a(T) − γ T4 , T, t ≥ 0, describes a multiple of Rin − Rout, and its very slow periodicity in t is initiated by the assumption on the solar constant Q(t) = Q0 + b sin(ωt). Let us compare this quantity, sketched in Figure 1.9 schematically for two times, say t1, t2 such that Q takes its minimum at t1 and its maximum at t2. The graph of f moves periodically between the two extreme positions. Note that in the one-dimensional situation considered, f(t, ·) can be seen as the negative gradient of a potential function U(t, ·) which depends periodically on time t.
- 28. 1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 7 0 0 f(t2, T) f(t1, T) T T T1(t1) T3(t1) T2(t1) T3(t2) T2(t2) T1(t2) Figure 1.9. Schematical form of radiation power difference at times t1 and t2 Figure 1.10. Potential function U at time instants t1 and t2. We now turn to a more general context. For simplicity of the heuristical exposi- tion still sticking to a one-dimensional scenario, we start by considering a temporally varying potential function U and set f(t, ·) = − ∂ ∂x U(t, ·), t ≥ 0. We assume that U oscillates in time between the two extreme positions depicted schematically in Figure 1.10. In Figure 1.10 (l.), the potential well on left hand side is deeper than on the right hand side, in Figure 1.10 (r.) the role of the deeper well has changed. As t varies, we will observe a smoothly time dependent potential with two wells of periodically and smoothly fluctuating relative depth. Just the function describing the position of the deepest well will in general be discontinuous. It will play a crucial role in the analysis now sketched. We assume in the sequel for simplicity that U(t, x), t ≥ 0, x ∈ R, is a smooth function such that for all t ≥ 0, U(t, ·) has exactly two minima, one at −1, the other at 1, and that the two wells at −1 and 1 are separated by the saddle 0, where U(t, 0) is assumed to take the value 0. Two moment pictures of the potential may look as in Figure 1.10. We further assume time periodicity for U, more formally that U(t, ·) = U(t + 1, ·).
- 29. 8 1. HEURISTICS The variable period of the input will be denoted by some positive number T. We therefore consider the stochastic differential equation (1.3) d dt Xε t = f t T , Xε t + √ ε Ẇt, with a one-dimensional Wiener process W (white noise Ẇ). We may circumscribe a more mathematical concept of stochastic resonance like this: given T (ω = 1 T ), find the parameter ε = ε(T) such that Xε is optimally tuned with the periodic input f( t T , ·). We pose the problem in the following (almost equivalent) way: given ε 0, find the good scale T = T(ε) such that optimal tuning of Xε with the periodic input is given, at least in the limit ε → 0. 1.2.1. Random motion of a strongly damped Brownian particle. The analogy with the motion of a physical particle in a periodically changing double well potential landscape alluded to in (1.3) (see also Mazo [72] and Schweitzer [97]) motivates us to pause for a moment and give it a little more thought. As in the previous section, let us concentrate on a one-dimensional setting, remarking that our treatment easily generalizes to a finite-dimensional setting. Due to Newton’s law, the motion of a particle is governed by the impact of all forces acting on it. Let us denote F the sum of these forces, m the mass, x the space coordinate and v the velocity of the particle. Then mv̇ = F. Let us first assume the potential to be turned off. In their pioneering work at the turn of the twentieth century, Marian Smoluchowski and Paul Langevin introduced stochastic concepts to describe the Brownian particle motion by claiming that at time t F(t) = −γv(t) + 2kTγẆt. The first term results from friction γ and is velocity dependent. An additional sto- chastic force represents random interactions between Brownian particles and their simple molecular random environment. The white noise Ẇ (the formal derivative of a Wiener process) plays the crucial role. The diffusion coefficient (standard deviation of the random impact) is composed of Boltzmann’s constant k, friction and environmental temperature T. It satisfies the condition of the fluctuation- dissipation theorem expressing the balance of energy loss due to friction and energy gain resulting from noise. The equation of motion becomes ⎧ ⎨ ⎩ ẋ(t) = v(t), v̇(t) = − γ m v(t) + √ 2kTγ m Ẇt. In equilibrium, the stationary Ornstein–Uhlenbeck process provides its solution: v(t) = v(0) e− γ m t + √ 2kTγ m t 0 e− γ m (t−s) dWs. The ratio β := γ m determines the dynamic behavior. Let us focus on the over- damped situation with large friction and very small mass. Then for t 1 β = τ (relaxation time), the first term in the expression for velocity can be neglected, while the stochastic integral represents a Gaussian process. By integrating, we
- 30. 1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 9 obtain in the over-damped limit (β → ∞) that v and thus x is Gaussian with almost constant mean m(t) = x(0) + 1 − e−βt β v(0) ≈ x(0) and covariance close to the covariance of white noise, see Nelson [82]: K(s, t) = 2kT γ min(s, t) + kT γβ − 2 + 2e−βt + 2e−βs − e−β|t−s| − e−β(t+s) ≈ 2kT γ min(s, t), s, t ≥ 0. Hence the time-dependent change of the velocity of the Brownian particle can be neglected, the velocity rapidly converges to thermal equilibrium (v̇ ≈ 0), while the spatial coordinate remains far from it. In the so-called adiabatic transformation, the evolution of the particle’s position is thus given by the transformed Langevin equation ẋ(t) = √ 2kT γ Ẇt. Let us next suppose that we face a Brownian particle in an external field of force, associated with a potential U(t, x), t ≥ 0, x ∈ R. This then leads to the Langevin equation ⎧ ⎨ ⎩ ẋ(t) = v(t), mv̇(t) = −γ v(t) − ∂U ∂x (t, x(t)) + 2kTγ Ẇt. In the over-damped limit, after relaxation time, the adiabatic elimination of the fast variables (see Gardiner [46]) then leads to an equation similar to the one encountered in the previous section, namely ẋ(t) = − 1 γ ∂U ∂x (t, x(t)) + √ 2kT γ Ẇt. 1.2.2. Time independent potential. We now continue discussing the heuris- tics of stochastic resonance for systems described by equations of the type encoun- tered in the previous two sections. To motivate the link to the theory of large deviations, we first study the case in which U(t, ·) is given by some time inde- pendent potential function U for all t. Following Freidlin and Wentzell [40], the description of the asymptotics contained in the large deviations principle requires the crucial notion of action functional. It is defined for T 0 and absolutely continuous functions ϕ: [0, T] → R with derivative ϕ̇ by S0T (ϕ) = 1 2 T 0 ϕ̇s − − ∂ ∂x U (ϕs) 2 ds. By means of the action functional we can define the quasipotential function V (x, y) = inf{S0T (ϕ): ϕ0 = x, ϕT = y, T 0}, for x, y ∈ R. Intuitively, V (x, y) describes the minimal work to be done in the potential landscape given by U to pass from x to y. Keeping this in mind, the relationship between U and V is easy to understand (for a more formal argument see Chapter 3). If x and y are in the same potential well, we have (1.4) V (x, y) = 2(U(y) − U(x))+ ,
- 31. 10 1. HEURISTICS where a+ = a ∨ 0 = max{a, 0} denotes the positive part of a real number a. In particular, if U(y) U(x), then V (x, y) = 0, i.e. going downhill in the landscape does not require work. If, however, x and y are in different potential wells, we have (recall U(0) = 0) (1.5) V (x, y) = −2U(x). This equation reflects the fact that the minimal work to do to pass to y consists in reaching the saddle 0, since then one can just go downhill. Rudiments of the following arguments can also be found in the explanation of stochastic resonance by McNamara and Wiesenfeld [74]. The main ingredient is the exit time law by Freidlin and Wentzell [40] (see also Eyring [37], Kramers [65] and Bovier et al. [14]). For y ∈ R, ε 0 the first time y is visited is defined by τε y = inf{t ≥ 0: Xε t = y}. If Px denotes the law of the diffusion (Xε t )t≥0 started at x ∈ R, the exit time law states that for any δ 0, x ∈ R we have (1.6) Px e V (x,y)−δ ε ≤ τε y ≤ e V (x,y)+δ ε → 1 as ε → 0. In other words, in the limit ε → 0, the process started at x takes approximately time exp(V (x,y) ε ) to reach y, or more roughly ε ln τε y ∼ = V (x, y) as ε → 0. As a consequence, one finds that as ε → 0, on time scales T(ε) at least as long as exp(V (x,y) ε ) or such that ε ln T(ε) V (x, y), we may expect with Px-probability close to 1 that the process Xε tT (ε) has reached y by time 1. Remembering (1.4) and (1.5) one obtains the following statement formulated much more generally by Freidlin. Suppose (1.7) lim ε→0 ε ln T(ε) 2 max{−U(−1), −U(1)}, and U(−1) U(1). Then the Lebesgue measure of the set (1.8) t ∈ [0, 1]: |Xε tT (ε) − (−1)| δ tends to 0 in Px-probability as ε → 0, for any δ 0. In other words, the process Xε , run in a time scale T(ε) large enough, will spend most of the time in the deeper potential well. Excursions to the other well are exponentially negligible on this scale, as ε → 0. The picture is roughly as deployed in Figure 1.11. 1.2.3. Periodic step potentials and quasi-deterministic motion. As a rough approximation of temporally continuously varying potential functions we may consider periodic step function potentials such as (1.9) U(t, ·) = U1(·), t ∈ [k, k + 1 2 ), U2(·), t ∈ [k + 1 2 , k + 1), k ∈ N0. We assume that both U1 and U2 are of the type described above, that U1(x) = U2(−x), x ∈ R, and that U1 has a well of depth V 2 at −1, and a well of depth v 2 at
- 32. 1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 11 Figure 1.11. Solution trajectory of the diffusion Xε tT (ε) in the time independent double-well potential U. 1, with V v (and U2 wells with respectively opposite roles). Let us briefly point out the main features of the transition times for periodic step potentials described in (1.6). According to (1.6) the exponential rate of the transition time from −1 to 1 in U1 in the small noise limit is asymptotically given by exp(V ε ), as long as the time scale T of the diffusion allows no switching of the potential states before, i.e. as long as T = T(ε) exp(V ε ). Accordingly, the transition time from 1 to −1 in U1 is given by exp(v ε ), as long as T = T(ε) exp(v ε ). Similar statements hold for transitions between states of U2. It is therefore also plausible that (1.8) generalizes to the following statement due to Freidlin [39, Theorem 2]. Suppose (1.10) lim ε→0 ε ln T(ε) V. Define φ(t) = −1, t ∈ [k, k + 1 2 ), 1, t ∈ [k + 1 2 , k + 1), k ∈ N0. Then the Lebesgue measure of the set (1.11) t ∈ [0, 1]: |Xε t T (ε) − φ(t)| δ tends to 0 as ε → 0 in Px-probability, for any δ 0, x ∈ R. Again, this just means that the process Xε , run in a time scale T(ε) large enough, will spend most of the time in the minimum of the deepest potential well which is given by the time periodic function φ. Excursions to the other well are exponentially negligible on this scale, as ε → 0. The picture is typically the one depicted in Figure 1.12. 1.2.4. Periodic potentials and quasi-deterministic motion. Since the function φ appearing in the previous theorem is already discontinuous, it is plausible that the step function potential is in fact a reasonable approximation of the general case of continuously and (slowly) periodically changing potential functions. It is intuitively clear how the result has to be generalized to this situation. We just have to replace the periodic step potentials by potentials frozen along a partition of the period interval on the potential state taken at its starting point, and finally let the mesh of the partition tend to 0. To continue the discussion in the spirit of the previous section and with the idea of instantaneously frozen potential states, we
- 33. 12 1. HEURISTICS Figure 1.12. Solution trajectory of the diffusion Xε tT (ε) in the double-well periodic step potential. have to explain the asymptotics of the minimal time a Brownian particle needs to exit from the (frozen) starting well, say the left one. Freezing the potential at some time s, the asymptotics of its exit time is derived from the classical large deviation theory of randomly perturbed dynamical systems, see Freidlin and Wentzell [40]. Let us assume that U is locally Lipschitz continuous. We recall that for any t ≥ 0 the potential U(t, ·) has its minima at −1 and 1, separated by the saddle point 0. The law of the first exit time τε 1 = inf{t ≥ 0: Xε t 0} is described by some particular functional related to large deviation. For t 0, we introduce the action functional on the space of real valued continuous functions C([0, t], R) on [0, t] by Ss t (ϕ) = ⎧ ⎨ ⎩ 1 2 t 0 ϕ̇u + ∂ ∂x U(s, ϕu) 2 du if ϕ is absolutely continuous, +∞ otherwise, which is non-negative and vanishes on the set of solutions of the ordinary differential equation ϕ̇ = − ∂ ∂x U(s, ϕ). Let x and y be real numbers. With respect to the (frozen) action functional, we define the (frozen) quasipotential Vs(x, y) = inf{Ss t (ϕ): ϕ ∈ C([0, t], R), ϕ0 = x, ϕt = y, t ≥ 0} which represents the minimal work the diffusion with a potential frozen at time s and starting in x has to do in order to reach y. To switch wells, the Brownian particle starting in the left well’s bottom −1 has to overcome the barrier. So we let V s = Vs(−1, 0). This minimal work needed to exit from the left well can be computed explicitly, and is equal to twice its depth at time s. The asymptotic behavior of the exit time is expressed by lim ε→0 ε ln Exτε 1 = V s or in generalization of (1.6) lim ε→0 Px e V s−δ ε τε 1 e V s+δ ε = 1 for any δ 0 and x 0. Let us now assume that the left well is the deeper one at time s. If the Brownian particle has enough time to cross the barrier, i.e. if T(ε) e V s ε , then, generalizing
- 34. 1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 13 (1.8), Freidlin in [39, Theorem 1] proves that independently of the starting point x it should stay near −1 in the following sense. The Lebesgue measure of the set t ∈ [0, 1]: |Xε tT (ε) − (−1)| δ converges to 0 in probability as ε → 0. If T(ε) e V s ε , the time left is not long enough for any crossing: the particle, starting at x, stays in the starting well, near the stable equilibrium point. In other words, the Lebesgue measure of the set t ∈ [0, 1]: |Xε tT (ε) − (−I(−∞,0)(Xε tT (ε)) + I[0,∞)(Xε tT (ε)))| δ converges to 0 in the small noise limit. This observation is at the basis of Freidlin’s law of quasi-deterministic periodic motion discussed in the subsequent section. The lesson it teaches is this: to observe switching of the position to the energetically most favorable well, T(ε) should be larger than some critical level e λ ε , where λ = infs≥0 V s. Measuring time in exponential scales by μ through the equation T(ε) = e μ ε , the condition translates into μ λ. Continuing the reasoning of the preceding subsection, if this condition is satisfied, we may define a periodic function φ denoting the deepest well position in dependence on t. Then, in generalization of (1.11), the process Xε will spend most of the time, measured by Lebesgue’s measure, near φ for small ε. 1.2.5. Quality of periodic tuning and reduced motion. Do the mani- festations of quasi-deterministic motion in instantaneously frozen potentials just discussed explain stochastic resonance? The problem is obvious. They just give lower bounds for the scale T(ε) = e μ ε for which noise strength ε leads to random switches between the most probable potential wells near the (periodic) deterministic times when the role of the deepest well switches. But if μ is too big, occasional excursions into the higher well will destroy a truly periodic tuning with the po- tential (see Figure 1.12). Just the duration of the excursions, being exponentially smaller than the periods of dwelling in the deeper well, will not be noticed by the residence time criteria discussed. We therefore also need an upper bound for possible scales. In order to find this optimal tuning scale μR λ, we first have to measure goodness of periodic tuning of the trajectories of the solution. In the huge physics literature on stochastic resonance, two families of criteria can be dis- tinguished. The first one is based on invariant measures and spectral properties of the infinitesimal generator associated with the diffusion Xε . Now, Xε is not time autonomous and consequently does not admit invariant measures. By taking into account deterministic motion of time in the interval of periodicity and considering the time autonomous process Zε t = (t mod T(ε), Xε t ), t ≥ 0, we obtain a Markov process with an invariant measure νε t (x) dt dx. In particular, for t ≥ 0 the law of Xε t ∼ νε t (x) dx and the law of Xε t+T (ε) ∼ νε t+T (ε)(x) dx, under this measure are the same for all t ≥ 0. Let us present the most important notions of quality of tuning (see Jung [62], or Gammaitoni et al. [43]): • the spectral power amplification (SPA) which plays an eminent role in the physics literature and describes the energy carried by the spectral component of the averaged trajectories of Xε corresponding to the period of the signal: ηX (ε, T) = 1 0 EνXε sT · e2πis ds 2 , ε 0, T 0.
- 35. 14 1. HEURISTICS • the total energy of the averaged trajectories EnX (ε, T) = 1 0 EνXsT 2 ds, ε 0, T 0. The second family of criteria is more probabilistic. It refers to quality measures purely based on the location of transition times between domains of attraction of the local minima, and residence time distributions measuring the time spent in one well between two transitions, or interspike times. This family, to be discussed in more detail in Section 1.4 below is certainly less popular in the physics community. As will turn out later, these physical notions of quality of periodic tuning of random trajectories exhibit one important drawback: they are not robust with respect to model resolution. It is here that an important concept of model reduction enters the stage. It is based on the conjecture that the effective dynamical properties of periodically forced diffusion processes as given by (1.3) can be traced back to finite state Markov chains periodically hopping between the stable equilibria of the potential function underlying the diffusion, for which the smallness parameter of the noise intensity is simply reflected in the transition matrix. These Markov chains should be designed to capture the essential information about the inter-well dynamics of the diffusion, while intra-well small fluctuations of the diffusion near the potential minima are neglected. Investigating goodness of tuning according to the different physical measures of quality makes sense both for the Markov chains as for the diffusions. This idea of model reduction was captured and followed in the physics literature in Eckmann and Thomas [32], McNamara and Wiesenfeld [74], and Nicolis [83]. In fact, theoretical work on the concept of stochastic resonance in the physics literature is based on the model reduction approach, see the surveys Anishchenko et al. [1], Gammaitoni et al. [43, 44], Moss et al. [79], and Wellens et al. [108]. As we shall see in Chapter 3, the optimal tuning relations between ε and T do not necessarily agree for Markov chains and diffusions. Even in the small noise limit discrepancies may persist that are caused by very subtle geometric properties of the potential function. It is our goal to present a notion of quality of periodic tuning which possesses this robustness property when passing from the Markov chains capturing the effective dynamics to the original diffusions. For this reason we shall study the different physical notions of quality of tuning first in the context of typical finite state Markov chains with periodically forced transition matrices. 1.3. Markov chains for the effective dynamics and the physical paradigm of spectral power amplification To keep this heuristic exposition of the main ideas of our mathematical ap- proach as simple as possible, besides allowing only two states for our Markov chain that play the role of the stable equilibria of the potential −1 and 1, let us also dis- cretize time. We continue to assume as in the discussion of periodically switching potential states above that U1(−1) = U2(1) = −V 2 , and U1(1) = U2(−1) = −v 2 . In a setting better adapted to our continuous time diffusion processes, in Chapter 3 time continuous Markov chains switching between two states will capture the effective diffusion dynamics. Hence, we follow here Pavlyukevich [86] and Imkeller and Pavlyukevich [58] and shall assume in this section that the parameter T in our model describing the period length, is an even integer. So for T ∈ 2N, ε 0,
- 36. 1.3. EFFECTIVE DYNAMICS VIA DICRETE MARKOV CHAINS 15 consider a Markov chain Y ε = (Y ε (k))k≥0 on the state space S = {−1, 1}. Let PT (k) be the matrix of one-step transition probabilities at time k. If we denote p− T (k) = P(Y ε (k) = −1), p+ T (k) = P(Y ε (k) = 1), and write P∗ for the transposed matrix, we have p− T (k + 1) p+ T (k + 1) = P∗ T (k) p− T (k) p+ T (k) . In order to model the periodic switching of the double-well potential in our Markov chains, we define the transition matrix PT to be periodic in time with period T. More precisely, PT (k) = Q1, 0 ≤ k mod T ≤ T 2 − 1, Q2, T 2 ≤ k mod T ≤ T − 1, with (1.12) Q1 = 1 − ϕ ϕ ψ 1 − ψ , Q2 = 1 − ψ ψ ϕ 1 − ϕ , ϕ = pe−V/ε , ψ = qe−v/ε , where 0 ≤ p, q ≤ 1, 0 v V +∞, 0 ε ∞. The entries of the transition matrices clearly are designed to mimic transition rates between −1 and 1 or vice versa that correspond to the transition times of the diffusion processes between the meta-stable equilibria, given according to the preceding section by exp(V ε ) resp. exp(v ε ). The exponential factors in the one-step transition probabilities are just chosen to be the inverses of those mean transition times. This is exactly what elementary Markov chain theory requires in equilibrium. The phenomenological prefactors p and q, chosen between 0 and 1, add asymmetry to the picture. It is well known that for a time-homogeneous Markov chain on {−1, 1} with transition matrix PT one can talk about equilibrium, given by the stationary distri- bution, to which the law of the chain converges exponentially fast. The stationary distribution can be found by solving the matrix equation π = P∗ T π with norming condition π− + π+ = 1. For time non homogeneous Markov chains with time periodic transition matrix, the situation is quite similar. Enlarging the state space S to ST = {−1, 1} × {0, 1, . . . , T − 1}, we recover a time homogeneous chain by setting Zε (k) = (Y ε (k), k mod T), k ≥ 0, to which the previous remarks apply. For convenience of notation, we assume ST to be ordered in the following way: ST = (−1, 0), (1, 0), (−1, 1), (1, 1), . . ., (−1, T − 1), (1, T − 1) . Writing AT for the matrix of one-step transition probabilities of Zε , the station- ary distribution R = (r(i, j))∗ is obtained as a normalized solution of the matrix equation (A∗ T − E)R = 0, E being the identity matrix. We shall be dealing with the following variant of stationary measure, which is not normalized in time. Let πT (k) = (π− T (k), π+ T (k))∗ = (r(−1, k), r(1, k))∗ , 0 ≤ k ≤ T − 1. We call the family πT = (πT (k))0≤k≤T −1 the stationary distribution of the Markov chain Y ε .
- 37. 16 1. HEURISTICS The matrix AT of one-step transition probabilities of Zε is explicitly given by AT = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 Q1 0 0 · · · 0 0 0 0 0 Q1 0 · · · 0 0 0 . . . . . . . . . . . . ... . . . . . . . . . 0 0 0 0 · · · 0 Q2 0 0 0 0 0 · · · 0 0 Q2 Q2 0 0 0 · · · 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . The matrix AT has block structure. In this notation 0 means a 2×2-matrix with all entries equal to zero, Q1, and Q2 are the 2-dimensional matrices defined in (1.12). Applying some algebra we see that the equation (A∗ T − E)R = 0 is equivalent to A T R = 0, where A T = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Q − E 0 0 0 · · · 0 0 0 Q∗ 1 −E 0 0 · · · 0 0 0 . . . . . . . . . . . . ... . . . . . . . . . 0 0 0 0 · · · −E 0 0 0 0 0 0 · · · P∗ 2 −E 0 0 0 0 0 · · · 0 Q∗ 2 −E ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ and Q = Q∗ 2Q∗ 2 · · · Q∗ 1Q∗ 1 = (Q∗ 2) T 2 (Q∗ 1) T 2 . But A T is a block-wise lower diagonal matrix, and so A T R = 0 can be solved in the usual way resulting in the following formulas. For every T ∈ 2N, the stationary distribution πT of Y ε with matrices of one- step probabilities defined in (1.12) is given by (1.13) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ π− T (l) = ψ ϕ + ψ + ϕ − ψ ϕ + ψ · (1 − ϕ − ψ)l 1 + (1 − ϕ − ψ) T 2 , π+ T (l) = ϕ ϕ + ψ − ϕ − ψ ϕ + ψ · (1 − ϕ − ψ)l 1 + (1 − ϕ − ψ) T 2 , π− T (l + T 2 ) = π+ T (l), π+ T (l + T 2 ) = π− T (l), 0 ≤ l ≤ T 2 − 1. The proof of (1.13) is easy and instructive, and will be contained in the following arguments. Note that πT (0) satisfies the matrix equation (Q∗ 2) T 2 (Q∗ 1) T 2 − E πT (0) = 0 with additional condition π− T (0) + π+ T (0) = 1. To calculate (Q∗ 2) T 2 (Q∗ 1) T 2 , we use a formula for the k-th power of 2 × 2-matrices Q = 1 − a a b 1 − b , a, b ∈ R, proved in a straightforward way by induction on k which reads 1 − a a b 1 − b k = 1 a + b b a b a + (1 − a − b)k a + b a −a −b b .
- 38. 1.3. EFFECTIVE DYNAMICS VIA DICRETE MARKOV CHAINS 17 Using some more elementary algebra we find (Q∗ 2) T 2 (Q∗ 1) T 2 = (Q1) T 2 (Q2) T 2 ∗ = 1 − ψ ψ ϕ 1 − ϕ T 2 1 − ϕ ϕ ψ 1 − ψ T 2 = 1 ϕ + ψ ϕ ϕ ψ ψ + (1 − ϕ − ψ) T 2 ϕ − ψ ϕ + ψ −1 −1 1 1 + (1 − ϕ − ψ)T ϕ + ψ ϕ −ψ −ϕ ψ , from which another straightforward calculation yields ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ π− T (0) = ϕ ϕ + ψ + ψ ϕ + ψ · (1 − ϕ − ψ) T 2 1 + (1 − ϕ − ψ) T 2 , π+ T (0) = ψ ϕ + ψ + ϕ ϕ + ψ · (1 − ϕ − ψ) T 2 1 + (1 − ϕ − ψ) T 2 . To compute the remaining entries, we use πT (l) = (Q∗ 1)l πT (0) for 0 ≤ l ≤ T 2 − 1, and πT (l) = (Q∗ 2)l (Q∗ 1) T 2 πT (0) for T 2 ≤ l ≤ T − 1 to obtain (1.13). Note also the symmetry π− T (l + T 2 ) = π+ T (l) and π+ T (l + T 2 ) = π− T (l), 0 ≤ l ≤ T 2 − 1. To motivate the physical quality of tuning concept of spectral power amplifica- tion, we first remark that our Markov chain Y ε can be interpreted as amplifier of the periodic input signal of period T. In the stationary regime, i.e. if the law of Y ε is given by the measure πT , the power carried by the output Markov chain at frequency a/T is a random variable ξT (a) = 1 T T −1 l=0 Y ε (l)e 2πia T l . We define the spectral power amplification (SPA) as the relative expected power carried by the component of the output with (input) frequency 1 T . It is given by ηY (ε, T) = EπT ξT (1) 2 , ε 0, T ∈ 2N. Here EπT denotes expectation w.r.t. the stationary law πT . The explicit description of the invariant measure now readily yields an explicit formula for the spectral power amplification. In fact, using (1.13) one immediately gets EπT ξT (1) = 1 T T −1 k=0 EπT Y ε (k)e 2πi T k = 1 − eπi T T 2 −1 k=0 (π+ T (k) − π− T (k))e 2πi T k = 4 T ϕ − ψ ϕ + ψ 1 1 − e 2πi T − 1 1 − (1 − ϕ − ψ)e 2πi T . Elementary algebra then leads to the following description of the spectral power amplification coefficient of the Markov chain Y ε for ε 0, T ∈ 2N: (1.14) ηY (ε, T) = 4 T2 sin2 ( π T ) · (ϕ − ψ)2 (ϕ + ψ)2 + 4(1 − ϕ − ψ) sin2 ( π T ) . Note now that the one-step probabilities Q1 and Q2 depend on the parameters noise level ε. Our next goal is to tune this parameter to a value which maximizes
- 39. 18 1. HEURISTICS the amplification coefficient ηY (ε, T) as a function of ε. So the stochastic resonance point is marked by the maximum of the spectral power amplification coefficient as a function of ε. To calculate it, substitute e−1/ε = x, and differentiate the explicit formula (1.14). The resulting relationship between period length T(ε) and noise intensity ε marking the stochastic resonance point can be recast in the formula T(ε) ∼ = 1 2π pq V − v v exp V + v 2ε . The maximal value of spectral power amplification is given by lim ε→0 ηY (ε, T(ε)) = 4 π2 0 ε 1 2 ηY (ε, T) 4 π2 Figure 1.13. The coefficient of the spectral power amplification ε → ηY (ε, T) for p = q = 0.5, V = 2, v = 1, T = 10 000. We also see that the spectral power amplification as a measure of quality of sto- chastic resonance allows to distinguish a unique time scale, and find its exponential rate V +v 2 together with the pre-exponential factor. The optimal exponential rate is therefore given by the arithmetic mean of the two potential barriers marked by the deep and shallow well of our double well potential. This basic relationship will appear repeatedly at different stages of our mathematical elaboration of concepts of optimal tuning. We may summarize our findings so far for discrete Markov chains that capture the effective dynamics of the potential diffusions which are our main subject of in- terest. Following the physics literature (e.g. Gammaitoni et al. [43] and McNamara and Wiesenfeld [74]) we understand stochastic resonance as optimal spectral power amplification. The closely related notion of signal-to-noise ratio and other reason- able concepts based on quality measures such as the relative entropy of invariant laws are discussed for Markov chains in Chapter 3 (see Section 3.2). The spectral power amplification coefficient measures the power carried by the expected Fourier coefficient in equilibrium of the Markov chain switching between the stable equilib- ria of the potential landscape of the diffusion which corresponds to the frequency of the underlying periodic deterministic signal. 1.4. Diffusions with continuously varying potentials The concept of spectral power amplification is readily extended to Markov chains in continuous time, still designed to capture the effective diffusion dynamics
- 40. 1.4. DIFFUSIONS WITH CONTINUOUSLY VARYING POTENTIALS 19 in higher dimensions, as well as to potential diffusions themselves. This will be done in detail in Chapter 3. However, it will turn out that diffusions and their reduced dynamics Markov chains are not as similar as expected. Indeed, in a reasonably large time window around the resonance point for Y ε , the tuning picture of the spectral power amplification for the diffusion is different. Under weak regularity conditions on the potential, it exhibits strict monotonicity in the window. Hence optimal tuning points for diffusion and Markov chain differ essentially. In other words, the diffusion’s SPA tuning behavior is not robust for passage to the reduced model (see Chapter 3, subsection 3.4.4). This strange deficiency is difficult to explain. The main reason of this subtle effect appears to be that the diffusive nature of the Brownian particle is neglected in the reduced model. In order to point out this feature, we may compute the SPA coefficient of g(Xε ) where g is a particular function designed to cut out the small fluctuations of the diffusion in the neighborhood of the bottoms of the wells, by identifying all states there. So g(x) = −1 (resp. 1) in some neighborhood of −1 (resp. 1) and otherwise g is the identity. This results in η̃X (ε, T) = 1 0 Eνg(Xε sT ) e2πis ds 2 , ε 0, T ≥ 0. In the small noise limit this quality function admits a local maximum close to the resonance point of the reduced model: the growth rate of Topt(ε) is also given by the arithmetic mean of the wells’ depths. So the lack of robustness seems to be due to the small fluctuations of the particle in the wells’ bottoms. In any case, this clearly calls for other quality measures to be used to transfer properties of the reduced model to the original one. Our discussion indicates that due to their emphasis on the pure transition dynamics, a second more probabilis- tic family of quality measures should be used. This will be made mathematically rigorous in Chapter 4. The family is composed of quality measures based on tran- sition times between the domains of attraction of the local minima, residence times distributions measuring the time spent in one well between two transitions, or inter- spike times. To explain its main features there is no need to restrict to landscapes frozen in time independent potential states on half period intervals. So from now on the potential U(t, x) is a continuous function in (t, x). For simplicity — remain- ing in the one-dimensional case — we further suppose that its local minima are given by ±1, and its only saddle point by 0, independently of time. So the only meta-stable states on the time axis are ±1. Let us denote by v−(t) 2 (resp. v+(t) 2 the depth of the left (resp. right) well. These function are continuous and 1-periodic. We shall assume that they are strictly monotonous between their global extrema. Let us now consider the motion of the Brownian particle in this landscape. As in the preceding case, according to Freidlin’s law of quasi-deterministic motion its trajectory gets close to the global minimum, if the period is large enough. The exponential rate of the period should be large enough to permit transitions: if T(ε) = eμ/ε with μ ≥ maxi=± supt≥0 vi(t) meaning that μ is larger than the max- imal work needed to cross the barrier, then the particle often switches between the two wells and should stay close to the deepest position in the landscape. By defining φ(t) = 2I{v+(t)v−(t)} − 1, in the small noise limit the Lebesgue measure of the set {t ∈ [0, 1]: |XtT − φ(t)| δ}
- 41. 20 1. HEURISTICS converges to 0 in probability for any δ 0. But in this case many transitions occur in practice, and the trajectory looks chaotical instead of periodic. So we have to choose smaller periods even if we cannot assure that the particle stays close to the global minimum since it needs some time to cross the barrier. Let us study the transition times. For this we assume that the starting point is −1 corresponding to the bottom of the deepest well. If the depth of the well is always larger than μ = ε ln T(ε), then the particle does not have enough time during one period to climb the barrier and should therefore stay in the starting well. On the contrary if the depth of the starting well becomes smaller than μ, the transition can and will happen. More precisely, for μ ∈ (inft≥0 v−(t), supt≥0 v−(t)) we define a− μ (s) = inf{t ≥ s: v−(t) ≤ μ}. The first transition time from −1 to 1 denoted τ+ has the following asymptotic behavior in the small noise limit: τ+/T(ε) → a− μ (0). The second transition which lets the particle return to the starting well will appear near the deterministic time a+ μ (a− μ (s))T(ε). The definitions of the coefficients a− μ and a+ μ are similar, the depth of the left well just being replaced by that of the right well. In order to observe periodic behavior of the trajectory, the particle has to stay a little time in the right well before going back. This will happen under the assumption v+(aμ(0)) μ, that is, the right well is the deepest one at the transition time. In fact we can then define the resonance interval IR, the set of all values μ such that the trajectories look periodic in the small noise limit: IR = max i=± inf t≥0 vi(t), inf t≥0 max i=± vi(t) . On this interval trajectories approach some deterministic periodic limit. We now outline the construction of a quality measure that is based on these observations, to be optimized in order to obtain stochastic resonance as the best possible response to periodic forcing. The measure we consider is based on the probability that a random transition of the diffusion happens during a small time window around the limiting deterministic transition time. Recall the transition times τε ±1 of Xε to ±1. For h 0, ε 0, T ≥ 0 let Mh (ε, T) = min i=± Pi τε ∓1 T(ε) ∈ [ai μ − h, ai μ + h] . In the small noise limit, this quality measure tends to 1 and optimal tuning can be obtained due to its asymptotic behavior described by the formula lim ε→0 ε ln(1 − Mh (ε, T)) = max i=± {μ − vi(ai μ − h)} for μ ∈ IR, uniformly on each compact subset. This property results from classical large deviation techniques applied to an approximation of the diffusion which is supposed to be locally time homogeneous, and will be derived in Chapter 4. Now we minimize the term on the left hand side in the preceding equality. In fact, if the window length 2h is small then μ − vi(ai μ − h) ≈ 2hv i(ai μ) since vi(ai μ) = μ by definition. The value v i(ai μ) is of course negative. Thus the position in which its absolute value is maximal should be identified. At this position the depth of the starting well drops most rapidly below the level μ. It is clear that for h small the eventually existing global minimizer μR(h) is a good candidate for the resonance point. To get rid of the dependence on h, we shall consider the limit of μR(h) as h → 0 denote by μR. This limit, if it exists, is called
- 42. 1.5. STOCHASTIC RESONANCE IN MODELS FROM ELECTRONICS TO BIOLOGY 21 the resonance point of the diffusion with time periodic landscape U. Let us note that for v−(t) = V +v 4 + V −v 4 cos(2πt) and v+(t) = v−(t + π), which corresponds to the case of periodically switching wells’ depths between v 2 to V 2 as in the frozen landscape case described above. Then the optimal tuning is T(ε) = exp(μR ε ) with μR = v+V 2 . This optimal rate is equivalent to the optimal rate given by the SPA coefficient. The big advantage of the quality measure based on the transition times is its robustness. Let us therefore consider the reduced model consisting in a two-state Markov chain with the infinitesimal generator Q(t) = −ϕ(t) ϕ(t) ψ(t) −ψ(t) , where ϕ(t) = exp(−v−(t/T ) ε ) and ψ(t) = exp(−v+(t/T ) ε ). The distribution of transi- tion times of this Markov chain is well known (see Chapter 4) and, divided by the period length, converges to ai μ. The reduced dynamics of the diffusion is captured by the Markov chain, and the optimization of the quality measure Mh (ε, T) for the Markov chain and the diffusion leads to the same resonance points. Our investigation focuses essentially on two criteria: one concerning the family of spectral measures, especially the spectral power amplification coefficient, and the other one dealing with transitions between the local minima of the potential. Many other criteria for optimal tuning between weak periodic signals in dynamical systems and stochastic response can be employed (see Chapter 3). The relation between long deterministic periods and noise intensity usually is expressed in exponential form T(ε) = exp(μ ε ), since the particle needs exponentially large times to cross the barrier separating the wells. This approach relies on the basic assumption that the barrier height is bounded below uniformly in time. This assumption which seems natural in the simple energy balance model of climate dynamics may be questionable in other situations. If the barrier height becomes small periodically on a scale related to the noise intensity, the Brownian particle does not need to wait an exponentially long time to climb it. In this scaling trajectories may appear periodic in the small noise limit. The modulation is assumed to be slow, but the time dependence does not have to be assumed exponentially slow in the noise intensity. In a series of papers [8, 9, 10, 11, 12] and in a monograph [13], Berglund and Gentz study the case in which the barrier between the wells becomes low twice per period: at time zero the right-hand well becomes almost flat and at the same time the bottom of the well and the saddle approach each other; half a period later, the scenario with the roles of the wells switched occurs. Even in this situation, there is a threshold value for the noise intensity under which transitions are unlikely and, above this threshold, trajectories typically exhibit two transitions per period. In this particular situation, optimal tuning can be described in terms of the concentration of sample paths in small space-time sets. 1.5. Stochastic resonance in models from electronics to biology As described in the preceding sections, the paradigm of stochastic resonance can quite generally and roughly be seen as the optimal amplification of a weak pe- riodic signal in a dynamical system triggered by random forcing. In this section, we shall briefly deviate from the presentation of our mathematical approach of optimal tuning by large deviations methods, illustrate the ubiquity of the phenomenon of
- 43. 22 1. HEURISTICS stochastic resonance. We will briefly discuss some prominent examples of dynami- cal systems arising in different areas of natural sciences in which it occurs, following several big reviews on stochastic resonance from the point of view of natural sciences such as [1, 43, 44, 79, 108]. We refer the reader to these references for ample fur- ther information on a huge number of examples where stochastic resonance appears. Finally we will briefly comment on computational aspects of stochastic resonance that are important in particular in high dimensional applications. 1.5.1. Resonant activation and Brownian ratchets. The two popular examples we mention here are elementary realizations of transition phenomena corresponding roughly to our paradigm of an overdamped Brownian particle in a potential landscape subject to weak periodic variation of some parameters. Here we face the examples of one-well potentials resp. asymmetric periodic multi-well potentials. The effect of the so-called resonant activation arises in the simple situation in which an overdamped Brownian particle exits from a single potential well with randomly fluctuating potential barrier. In the case we consider the potential barrier can be considered to undergo weak periodic deterministic fluctuations in contrast. Even in the simplest situation, in which the height of the potential barrier is given by a Markov chain switching between two states, one can observe a non-linear dependence of the mean first exit time from the potential well and the intensity of the switching (see e.g. Doering and Elston [28]). Noise induced transport in Brownian ratchets addresses the directed motion of the Brownian particle in a spatially asymmetric periodic potential having the shape of a long chain of downward directed sawtooths of equal length. It arises as another exit time phenomenon, since random exits over the lower potential barrier on the right hand side of the particle’s actual position are highly favored. For instance in the context of an electric conductor, this effect creates a current in the downward direction indicated, see Doering et al. [28, 27] and Reimann [91]. An important application of this effect is the biomolecular cargo transport, see e.g. Elston and Peskin [35] and Vanden–Eijnden et al. [80, 26]. 1.5.2. Threshold models and the Schmitt trigger. Models of stochastic resonance based on a bistable weakly periodic dynamical system of the type (0.1) are often referred to as dynamical models in contrast to the so-called non-dynamical or threshold models. These are models usually consisting of a biased deterministic input which may be periodic or not, and a multi-state output. In the simplest situation, the output takes a certain value as the input crosses a critical threshold. The simplest model of this type is the Schmitt trigger, an electronic device studied first by Fauve and Heslot [38] and Melnikov [76] (see also [1, 43, 69, 70, 74]). It is given by a well-known electronic circuit, characterized by a two-state output and a hysteretic loop. The circuit is supplied with the input voltage w = wt, which is an arbitrary function of time. In the ideal Schmitt trigger the output voltage Y = Yt has only two possible values, say −V and V . Let w increase from −∞. Then Y = V until w reaches the critical voltage level V+. As this happens, the output jumps instantaneously to the level −V . Decreasing w does not affect the output Y until w reaches the critical voltage V−. Then Y jumps back. Therefore, the Schmitt trigger is a bistable system with hysteresis, see Figure 1.14. The width of the hysteresis loop is V+ − V−. Applying a periodic voltage of small amplitude a
- 44. 1.5. STOCHASTIC RESONANCE IN MODELS FROM ELECTRONICS TO BIOLOGY 23 Figure 1.14. The input-output characteristic (hysteresis loop) of the Schmitt trigger. and period T 0, for example, to V+, we periodically modulate the critical level. After adding a random noise at the input, the system is able to jump between the two states ±V . As in the example of glacial cycles we can consider a discontinuous modulation, for instance given by V+(t) = a sign(sin (2πt T )). The whole picture is now similar to the one in (1.3). Here the periodic modulation of the reference voltage corresponds to the tilting of the potential wells. Fauve and Heslot [38] studied the power spectrum of the system and, as in the glacial cycle example, established that the energy carried by the spectral component of Y at a given driving frequency has a local maximum for a certain intensity of the input noise. The Schmitt trigger provides another interpretation to the phenomenon of sto- chastic resonance. A system displaying stochastic resonance can be considered as a random amplifier. The weak periodic signal which cannot be detected in the absence of noise, can be successfully recovered if the system (the Schmitt trigger or (1.3)) is appropriately tuned. In other words, the weak underlying periodicity is exhibited at appropriately chosen non-zero levels of noise, and gets lost if noise is either too small or too large. To date, the most important application area of threshold models is neural dynamics (see Bulsara et al. [17], Douglass et al. [29], Patel and Kosko [85]) and transmission of information (see Neiman et al. [81], Simonotto et al. [99], Stocks [101], Moss et al. [79]). The recent book [73] by McDonnell et al. gives a very complete account on the theory of non-dynamic or threshold stochastic resonance. 1.5.3. The paddlefish. In this well known and frequently discussed example stochastic resonance appears in the noise-enhanced feeding behavior of the pad- dlefish Polyodon spathula (see Greenwood et al. [47], Russel et al. [95], Freund et al. [42]). This species of fish lives in the Midwest of the United States and in the Yangtze River in China, and feeds on the zooplankton Daphnia. To detect its prey animals under limited visibility conditions at river bottoms, the paddlefish uses the long rostrum in front of its mouth as an electrosensory antenna. The frequency range of sensitivity of the rostrum’s electroreceptors well overlaps with the range of frequencies produced by the prey. Roughly, the capture probability is observed as a function of the position of the prey relative to the rostrum. In experiments, external noise was generated by electrodes connected to an electric noise genera- tor. It was observed that the spatial distribution and number of strike locations
- 45. 24 1. HEURISTICS is a function of the external noise intensity, with a maximum of captures of more distant plankton at some optimal external noise intensity. If experimentally noisy electric signals improve the sensitivity of the electroreceptors, nature itself should also provide sources of noise. In [95] it was conjectured that, besides the signal, such a noise might be produced by the populations of prey animals themselves. In [42] this conjecture was confirmed by measurements of the noise strength pro- duced by single Daphnia in the vicinity of a swarm. In the simple quantitative approaches, quality of tuning is measured by Fisher information, a concept that may be comparable to the entropy notions in Chapter 3, Section 3.2. 1.5.4. The FitzHugh–Nagumo system. A more detailed modeling of neu- ral activities of living systems underlies this well known and studied example. It deals with action potentials and electric currents transmitted through systems of ion channels provided by the axons in neural networks, triggered by their mutual interaction and the interaction of the system with the biological environment. Neu- rons communicate with each other or with muscle cells by means of electric signals. Each single neuron can be modeled as an excitable dynamical system: in the rest state characterized by a negative potential gap with respect to the extracellular environment, no current flows through the membrane of the neuronal cell. If this threshold potential barrier disappears due to noisy perturbations created by the environment (neighboring cells, external field), ion channels through the membrane are opened and currents appear in form of a spike or firing, followed by a deter- ministic recovery to the rest state. During a finite (refractory) time interval, the membrane potential is hyperpolarized by the current flow, and any firing impossi- ble. The theory that captures the above-mentioned features of neuronal dynamics, including the finite refractory time, is described by the FitzHugh–Nagumo (FHN) equations (see Kanamaru et al. [63]). In the diffusively coupled form, a system of N coupled neurons is described by the system of equations (see [63]) τu̇i(t) = − vi + ui − u3 i 3 + S(t) + √ ε Ẇi(t) + 1 N N j=1 (ui − uj), v̇i(t) = ui − βvi + γ. Here ui describes the membrane potential of neuron i, vi a variable describing whether and to which degree neuron i is in the refractory interval of time after firing, S describes an external periodic pulse acting on the potential levels, while W1, . . . , Wn is a vector of independent Brownian motions. Finally, τ, β, ε and γ are system parameters. In the infinite particle limit, the system becomes a sto- chastic partial differential equation. Roughly, total throughput current will be a function of the model parameters, and stochastic resonance appears as its optimal value for a suitable parameter choice (synchronization). The paper by Wiesenfeld et al. [109] reports about a much simplified form of this system, in which action potentials of single mechanoreceptor cells of the crayfish Procambarus clarkii are concerned. The mechanosensory system of the crayfish consists of hairs located on its tailfan, connected to mechanoreceptor cells. Streaming water moves the hair and so provides the external excitation that causes the mechanoreceptor cell to fire. Experimentally (see [109]), a piece of tailfan containing the hair and sensory neuron was extracted and put into a saline solution environment. Then, periodic pressure
- 46. 1.5. STOCHASTIC RESONANCE IN MODELS FROM ELECTRONICS TO BIOLOGY 25 modulations and random noise were imposed on the environment. The firings pro- duced by the mechanoreceptor cell were recorded for different noise levels, and show clear stochastic resonance peaks as functions of noise intensity. Similar phenomena are encountered on a much more general basis in the exchange of substances or information through ionic channels on cell membranes in living organisms. 1.5.5. Physiological systems. The fact that sensory neurons are excitable systems leading to the FitzHugh–Nagumo equations in the preceding subsection, is also basic for many suggestions of how to make use of the phenomenon of stochastic resonance in medicine. Disfunctions arising in sensory organs responsible for hear- ing, tactile or visual sensations or for balance control could result from relatively higher sensitivity thresholds compared with those of healthy organs. To raise the sensitivity level, a natural idea seems to be to apply the right amount of external noise to these dysfunctional organs, in order to let stochastic resonance effects am- plify weak signal responses. In experiments reported in Collins et al. [21], local indentations were applied to the tips of digits of test persons who had to correctly identify whether a stimulus was presented. Stimuli generated by subthreshold sig- nals garnished with noise led to improvements in correct identification, with some optimal noise level indicating a stochastic resonance point. Results like this may be used for designing practical devices, as for instance gloves, for individuals with ele- vated cutaneous sensory thresholds. Similarly, randomly vibrating shoe inserts may help restoring balance control (see Priplata et al. [89]). Stochastic resonance effects may be used for treating disfunctions of the human blood pressure system (barore- flex system) featuring a negative feedback between blood pressure and heart rate resp. width of blood vessels. Blood pressure is monitored by two types of receptors, for arteries and veins. In Hidaka [54], a weak periodic input was introduced at the venous blood pressure receptor, whereas noise was added to the arterial receptor. It was shown that the power of the output signal of the heart rate (measured by an electrocardiogram) as a function of noise intensity exhibits a bell-shape form, typical for a curve with a stochastic resonance point. Another group of possible medical applications of the amplification effects of stochastic resonance is related to the human brains information processing activity (see Mori and Kai [78]). In an experiment in Usher and Feingold [103] the effect of stochastic resonance in the speed of memory retrieval was exhibited. Test persons were proven to perform single digit calculations (e.g. 7 × 8 =?) significantly faster when exposed to an optimal level of acoustic noise (via headphones). 1.5.6. Optical systems. In optical systems, stochastic resonance was first observed in McNamara et al. [75] and Vemuri and Roy [104] in a bidirectional ring laser, i.e. a ring resonator with a dye as lasing medium. This laser system supports two meta-stable states realized as modes of the same frequency that travel in oppo- site directions. They are strongly coupled to each other by the lasing medium, thus permitting a bistable operation. When the pumping exceeds the lasing threshold, either clockwise or counterclockwise modes propagate in the laser, with switchings between those two modes initiated by spontaneous emission in the active medium, and fluctuations of the pump laser. The net gains of the two propagating modes in opposite directions can be controlled by an acousto-optical modulator inside the cavity, which thus can be used both to impose a periodic switching rate between the modes and to inject noise. Therefore the resulting semiclassical laser equations
- 47. 26 1. HEURISTICS are equivalent to those describing overdamped motion of a particle in a periodi- cally modulated double well potential, as described in the prototypical example of Section 1.2. The choice of examples we discussed in more detail is rather selective. The ef- fects of stochastic resonance have been found in a big number of dynamical systems in various further areas of the sciences, and studied by a variety of physical mea- sures of quality of tuning. We just mention a big field of applications in microscopic systems underlying the laws of quantum mechanics in which intrinsic quantum tun- neling effects interfere with the interpretation of potential barrier tunneling that can be seen as causing noise induced transitions in diffusion dynamics. See [108] for a comprehensive survey. Stochastic resonance has further been observed in passive optical bistable systems [30], in experiments with magnetoelastic ribbons [100], in chemical systems [67], as well as in further biological ones [94, 60, 41]. Stochastic resonance may even be observed in more general systems in which the role of periodic deterministic signals is taken by some other physical mechanisms (see [108]). 1.5.7. Computational aspects of large deviation theory related to stochastic resonance. Our theoretical approach aimed at explaining stochastic resonance conceptually by means of space-time large deviations of weakly periodic dynamical systems does not touch at all the field of numerical algorithms and sci- entific computing for stochastic resonance related quantities which become very important for applications especially in high dimensions. In the framework of the classical Freidlin–Wentzell theory, first exit time estimates as well as large devia- tions rates are analytically expressed by the quasi-potential (see Chapter 2) which can be calculated more or less explicitly for gradient systems. To determine and minimize the quasi-potential in high-dimensional scenarios is an analytically hardly accessible task. In Vanden–Eijnden et al. [31, 53] practically relevant algorithms with numerous applications for this task have been developed. They have been ap- plied to various problems in different areas of application of stochastic resonance.
- 48. CHAPTER 2 Transitions for time homogeneous dynamical systems with small noise The trigger of stochastic resonance are diffusion exits from domains, described by the exit times of randomly perturbed dynamical systems from domains of at- traction of their stable fixed points. These exits can be rigorously understood by means of large deviations for dynamical systems with small random perturbations, as has been shown in Freidlin and Wentzell [40]. The core ingredient of our ap- proach to stochastic resonance are therefore concepts of large deviations. In this chapter we present a self-contained treatment of the theory of large deviations for randomly perturbed dynamical systems with additive Gaussian noise. They are typically given by solutions of stochastic differential equations of the type (2.1) dXε t = b(Xε t ) dt + √ ε dWt, Xε 0 = x, with a smooth mapping b of Euclidean space Rd , and a d-dimensional Brownian motion W. To fix ideas in the framework of the most important paradigm for our treatment, think of b = −∇xU for a typically double well potential U on Rd . By the action of the Brownian noise, the two well bottoms become meta-stable, and we are primarily interested in describing the asymptotic exit and transition behavior of the trajectories of the solution of (2.1) between their domains of attraction, in the limit ε → 0. Asymptotic exit rates for these random times are derived via large deviations principles (LDP) for the laws με = P ◦ Xε of the diffusion process as ε → 0. In this case, με lives on the space of Rd -valued continuous functions defined on R+, endowed with the topology of uniform convergence on compact subintervals of R+. To derive these LDP, we start with the LDP for the Wiener process given by Schilder’s theorem. Since by our choice of noise, the solution of (2.1) depends continuously on Brownian motion, the LDP for W may be transferred to Xε via a contraction principle. To establish Schilder’s theorem from an elementary perspective, we use the Schauder representation of Brownian motion allowing an elegant approach according to Baldi and Roynette [3]. The derivation of the asymptotic properties of exit times in the small noise limit follows the classical treatment by Dembo and Zeitouni [25]. Here the famous Kramers– Eyring formula for the expected exit time is underpinned with mathematical rigor. In terms of the pseudopotential V associated with a dissipative drift b it states that the average time it takes to leave the domain of attraction of one of the two meta-stable equilibria, say x+, is given to exponential order by exp(V ε ). Here V = infy∈χ V (x+, y) is the critical cost to provide for leaving the domain, and χ is the manifold separating the basins of attraction. 27
- 49. 28 2. TIME HOMOGENEOUS SYSTEMS In Section 2.1 we start by representing one-dimensional Brownian motion in its Schauder decomposition (Theorem 2.5). It allows a direct approach to its regular- ity properties in terms of Hölder norms on spaces of continuous functions (Theo- rem 2.6). The key to this elegant and direct approach is Ciesielski’s isomorphism of normed spaces of continuous functions with sequence spaces via Fourier repre- sentation (Theorems 2.2 and 2.3). In Section 2.2 we recall general notions and basic concepts about large deviations, especially addressing their construction from exponential decay rates of probabilities of basis sets of topologies (Theorem 2.15), and their transport between different topological spaces via continuous mappings (contraction principle) (Theorems 2.17, 2.21). In Section 2.3 the elegant Schauder representation of Brownian motion then allows a derivation of Schilder’s large de- viation principle (LDP) for Brownian motion (Theorem 2.28) from the elementary LDP for one-dimensional Gaussian random variables (Theorem 2.22). The Freidlin– Wentzell theory, extended to locally Lipschitz diffusion coefficients, and culminating in the LDP for diffusion processes (Theorem 2.36) is presented in Section 2.4. Fi- nally, in Section 2.5 we follow Dembo and Zeitouni [25] to derive the exit time laws due to Freidlin and Wentzell [40] for time homogeneous diffusions from domains of attraction of underlying dynamical systems in the small noise limit (Theorem 2.42). 2.1. Brownian motion via Fourier series In this section, we shall present Brownian motion in an approach based on Fourier series with respect to the orthonormal system of Haar functions. This approach will be seen to open an easy and fast route to large deviations principles for Brownian motion, the basic noise process added to deterministic dynamical systems to provide the time homogeneous randomly perturbed dynamical systems that are the main objects of interest for this section. In fact, we shall present a direct proof of Schilder’s Theorem which only uses this Fourier series representation and the large deviation principle for one-dimensional Gaussian variables. The basic idea of this approach for large deviations on function spaces is triggered by an observation by Ciesielski according to which smoothness properties of functions in Hölder spaces can be studied via a universal Banach space isomorphism through convergence properties of sequences. We first present Ciesielski’s isomorphism. 2.1.1. The Ciesielski isomorphism of Hölder and sequence spaces. For 0 α ≤ 1, let Cα ([0, 1], R) be the space of all α-Hölder continuous functions f : [0, 1] → R starting at zero, f(0) = 0. This space is a Banach space endowed with the Hölder norm fα = sup 0≤ts≤1 |f(t) − f(s)| |t − s|α . Denote, moreover, by l∞ (N0, R) the Banach space of all bounded real valued se- quences η = (ηn)n≥0 endowed with the norm η∞ = supn≥0 |ηn|. We call two Banach spaces isomorphic if there exists a one-to-one linear map between the spaces. By means of Fourier decomposition of Hölder continuous functions with the Schauder basis we first prove that Cα ([0, 1], R) and l∞ (N0, R) are isomorphic.
- 50. 2.1. BROWNIAN MOTION VIA FOURIER SERIES 29 For this purpose we introduce the Haar functions. For t ∈ [0, 1] let χ0(t) ≡ 1 and χ2k+l(t) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ √ 2k if t ∈ 2l 2k+1 , 2l+1 2k+1 , − √ 2k if t ∈ 2l+1 2k+1 , 2l+2 2k+1 , k ≥ 0, l = 0, 1, . . . , 2k − 1, 0 otherwise. The family (χn)n≥0 is a complete orthonormal system of L2 ([0, T], Rd ) . The Schauder functions are just the primitives of the Haar system, given for t ∈ [0, 1] by φn(t) = t 0 χn(s) ds. If f ∈ Cα ([0, 1], R) possesses a square integrable density ˙ f ∈ L2 ([0, 1], R) so that for t ∈ [0, 1] we have f(t) = t 0 ˙ f(s) ds, we can write ˙ f = ∞ n=0 χn, ˙ fχn, and therefore f = ∞ n=0 χn, ˙ fφn. Indeed, due to the fact that for k ≥ 0 fixed, and 0 ≤ l1, l2 ≤ 2k − 1, l1 = l2, the supports of the functions φ2k+l1 and φ2k+l2 are disjoint and the functions are uniformly bounded by 2− k 2 −1 , we may estimate for K ∈ N, q ≥ p ≥ 2K by means of Cauchy–Schwarz’ inequality (2.2) q n=p χn, ˙ fφn ≤ ∞ k=K 2k −1 l=0 χ2k+l, ˙ fφ2k+l ≤ ∞ k=K sup 0≤l≤2k−1 |χ2k+l, ˙ f| · 2− k 2 −1 ≤ ∞ k=K 1 0 ˙ f2 (s) ds 1 2 · 2− k 2 −1 . This clearly implies the convergence of the series in the uniform norm. We shall now see by following Ciesielski [20] that this representation may be extended to Hölder spaces. For this purpose denote for n = 2k + l χn, ˙ f = √ 2k 2f 2l + 1 2k+1 − f 2l + 2 2k+1 − f 2l 2k+1 . This just gives the integral of the function χn with respect to f as an integrator. Lemma 2.1. Let 0 α ≤ 1, and let f ∈ Cα ([0, 1], R). Then f = ∞ n=0 χn, ˙ fφn, with convergence with respect to the uniform norm.
- 51. 30 2. TIME HOMOGENEOUS SYSTEMS Proof. It can be seen easily that f may be approximated in the uniform norm by a sequence (fm)m∈N of functions possessing square integrable densities ( ˙ fm)m∈N, and with α-Hölder norms bounded by the one of f. Take for instance a sequence obtained from f by smoothing with a sequence of smooth approximations of the unit. More precisely, let φ: [−1, 1] → R+ be a C∞ function such that 1 −1 φ(x) dx = 1. For m ∈ N, let φm(·) = mφ(m ·), and fm(t) = 1 −1 f(t − x)φm(x) dx, t ∈ [0, 1] (here we assume f to be trivially extended continuously to [−1, 2] by constant branches). Obviously, (fm)m∈N converges to f in the uniform norm, and for each m, fm possesses a square integrable density ˙ fm. Moreover, for s, t ∈ [0, 1], m ∈ N, α ∈ (0, 1], |fm(t) − fm(s)| |t − s|α ≤ 1 −1 |f(t − x) − f(s − x)| |t − s|α φm(x) dx, hence fmα ≤ fα. Since we know from the above discussion that the desired representations hold for fm for all m ∈ N, a dominated convergence argument shows that we have to prove (2.3) sup m∈N q n=p χn, ˙ fmφn → 0 as q ≥ p → ∞. To do this, we have to modify the estimate (2.2) a bit. In fact, for any m ∈ N, and K ∈ N, q ≥ p ≥ 2K we have (2.4) q n=p χn, ˙ fmφn ≤ ∞ k=K 2k −1 l=0 χ2k+l, ˙ fmφ2k+l ≤ ∞ k=K sup 0≤l≤2k−1 |χ2k+l, ˙ fm|2− k 2 −1 ≤ ∞ k=K fmα2−αk ≤ fα · ∞ k=K 2−αk . The latter expression obviously converges to 0 as K → ∞. This completes the proof. The following Theorem states that Cα ([0, 1], R) and l∞ (N0, R) are isomorphic. Theorem 2.2. Let 0 α 1. For n ∈ N0 let cn(α) = 1, n = 0, 2k(α− 1 2 )+α−1 , n = 2k + l, k ≥ 0, 0 ≤ l ≤ 2k − 1. Define Tα : Cα ([0, 1], R) → l∞ (N0, R), f → (cn(α) χn, ˙ f)n≥0. Then T−1 α : l∞ (N0, R) → Cα ([0, 1], R), (ηn)n≥0 → ∞ n=0 ηn cn(α) φn,
- 52. 2.1. BROWNIAN MOTION VIA FOURIER SERIES 31 Tα is an isomorphism, and for the operator norms we have the following inequalities: Tα = 1, T−1 α ≤ 2 (2α − 1)(21−α − 1) . Proof. By definition, for n = 2k + l, k ≥ 0, 0 ≤ l ≤ 2k − 1 we have (2.5) |χn, ˙ f| ≤ 2−(k+1)α+ k 2 +1 fα = 1 cn(α) fα. Therefore, Tα is well defined, and we have Tα ≤ 1. Moreover, for f(t) = t, 0 ≤ t ≤ 1, we have χ0, ˙ f = 1, while for n ∈ N we have χn, ˙ f = 0. Hence Tαf∞ = fα. This implies Tα = 1. Lemma 2.1 shows that Tα is one-to-one and that T−1 α is its inverse. We next prove the inequality for the operator norm of T−1 α . For η = (ηn)n≥0 ∈ l∞ (N0, R) given, choose 0 ≤ s t ≤ 1, and write f = T−1 α (η). Then we have (2.6) |f(t) − f(s)| ≤ η∞ · |t − s| + ∞ k=0 2k −1 l=0 1 c2k+l(α) |φ2k+l(t) − φ2k+l(s)| . Now choose k0 ≥ 0 such that 2−k0−1 |t − s| ≤ 2−k0 . Then for 0 ≤ k k0 due to the fact that the supports of φ2k+l, 0 ≤ l ≤ 2k − 1, are disjoint (2.7) 2k −1 l=0 1 c2k+l(α) |φ2k+l(t) − φ2k+l(s)| = sup 0≤l≤2k−1 1 c2k+l(α) |φ2k+l(t) − φ2k+l(s)| ≤ 2−k(α− 1 2 )−α+1 · 2 k 2 · |t − s| ≤ 2k(1−α)−α+1−k0(1−α) · |t − s|α = 2(1−α)(1+k−k0) · |t − s|α , while for k ≥ k0 (2.8) 2k −1 l=0 1 c2k+l(α) |φ2k+l(t) − φ2k+l(s)| ≤ 2−k(α− 1 2 )−α+1 · 2− k 2 ≤ 2−kα−α+1+(k0+1)α · |t − s|α = 2α(k0−k) · |t − s|α . Combining (2.6), (2.7) and (2.8), we obtain the estimate |f(t) − f(s)| |t − s|α ≤ 2 (2α − 1)(21−α − 1) η∞, and therefore T−1 α ≤ 2 (2α − 1)(21−α − 1) .
- 53. 32 2. TIME HOMOGENEOUS SYSTEMS The spaces we are ultimately interested in are those in which almost all sample paths of the Brownian motion are living. We therefore have to extend the isomor- phism of Theorem 2.2 to the following subspaces of Hölder continuous functions. For 0 α ≤ 1, let Cα 0 ([0, 1], R) be the subspace of Cα ([0, 1], R) composed of all functions f for which f(0) = 0 and lim δ→0 sup 0≤st≤1, |t−s|≤δ |f(t) − f(s)| |t − s|α = 0. The isomorphism of Theorem 2.2 will then be restricted to the subspace c0(N0, R) of all sequences η = (ηn)n≥0 in l∞ (N0, R) which converge to 0 as n → ∞ as a target space. The following Theorem holds. Theorem 2.3. Let 0 α 1. Let (cn(α))n≥0 be defined as in Theorem 2.2. Define Tα,0 : Cα 0 ([0, 1], R) → c0(N0, R), f → (cn(α) χn, ˙ f)n≥0. Then T−1 α,0 : c0(N0, R) → Cα 0 ([0, 1], R), (ηn)n≥0 → ∞ n=0 ηn cn(α) φn. The mapping Tα,0 is an isomorphism, and for the operator norms we have the following inequalities Tα,0 = 1, T−1 α,0 ≤ 2 (2α − 1)(21−α − 1) . Proof. Note first that (2.5) can be strengthened by definition to read (2.9) |χn, ˙ f| ≤ 1 cn(α) sup 0≤st≤1, |t−s|≤2−k−1 |f(t) − f(s)| |t − s|α . Hence, by definition of Cα 0 ([0, 1], R), we obviously obtain that Tα,0 is well defined. To prove that also T−1 α,0 is well defined, we just have a closer inspection of the arguments that led to the operator norm inequality in the proof of Theorem 2.2. First, note that for η = (ηn)n≥0 ∈ c0(N0, R), with f = T−1 α,0(η) (2.10) |f(t)−f(s)| ≤ |η0(α)|·|t−s|+ ∞ k=0 2k −1 l=0 1 c2k+l(α) |η2k+l|·|φ2k+l(t)−φ2k+l(s)|. Now choose again k0 ≥ 0 such that 2−k0−1 |t − s| ≤ 2−k0 , and denote τn = supk≥n |ηk|. Then for 0 ≤ k k0 (2.11) 2k −1 l=0 1 c2k+l(α) |η2k+l| · |φ2k+l(t) − φ2k+l(s)| = sup 0≤l≤2k−1 1 c2k+l(α) |η2k+l| · |φ2k+l(t) − φ2k+l(s)| ≤ τ2k · 2−k(α− 1 2 )−α+1 · 2 k 2 |t − s| ≤ τ2k · 2k(1−α)−α+1−k0(1−α) · |t − s|α = τ2k · (21−α )(1+k−k0) · |t − s|α ,
- 54. 2.1. BROWNIAN MOTION VIA FOURIER SERIES 33 while for k ≥ k0 (2.12) 2k −1 l=0 1 c2k+l(α) |η2k+l| · |φ2k+l(t) − φ2k+l(s)| ≤ τ2k0 · 2−k(α− 1 2 )−α+1 · 2− k 2 ≤ τ2k0 · 2−kα−α+1+(k0+1)α · |t − s|α = τ2k0 · (2α )(k0−k) · |t − s|α . Hence (2.10), (2.11) and (2.12) imply |f(t) − f(s)| |t − s|α ≤ η∞ · |t − s|1−α + k≤k0 τ2k · (21−α )(1+k−k0) + 1 2α − 1 τ2k0 . Now k0 → ∞ as |t − s| → 0. This, however, entails that f ∈ Cα 0 ([0, 1], R). All arguments used in the proof of Theorem 2.2 to show the inequalities for the operator norms are valid here. Just note that the function f(t) = t, 0 ≤ t ≤ 1, belongs to Cα 0 ([0, 1], R). 2.1.2. The Schauder representation of Brownian motion. We shall now present an approach to the study of one-dimensional Brownian motion which is close to Wiener’s representation of Brownian motion by Fourier series with trigonometric functions as a basis. Our basis will be given by the Haar functions and their primitives. In fact, the trajectories of Brownian motion will be described just as in the preceding section continuous functions were isomorphically described by sequences. Given a Brownian motion W indexed by the unit interval, with the same notation as in the preceding section we write it sample by sample as a series with coefficients χn, Ẇ, n ∈ N0. Due to the scaling properties and the structure of Haar functions, these random coefficients are i.i.d. standard normal random variables. This, in turn, allows us to construct Brownian motion indexed by the unit interval by taking any sequence of i.i.d. standard normal variables (Zn)n∈N0 on a probability space (Ω, F, P), and defining the stochastic process (2.13) Wt = ∞ n=0 Znφn(t), t ∈ [0, 1]. To get information about the quality of convergence of this Fourier series, we need to control the size of the random sequence (Zn)n∈N0 in the following Lemma. Lemma 2.4. Let (Zn)n≥2 be a sequence of standard gaussian random variables. There exists a real valued random variable C such that the inequality (2.14) sup n≥2 |Zn(ω)| ≤ C(ω) √ ln n holds true P-a.s. Proof. For x ≥ 1 and n ≥ 2 we have P(|Zn| ≥ x) = 2 π ∞ x e− u2 2 du ≤ 2 π ∞ x ue− u2 2 du = 2 π e− x2 2 . Hence for any β 1 P |Zn| ≥ 2β ln n ≤ 2 π e−β ln n = 2 π 1 nβ .
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- 56. may be used against them with even greater force. They talk of the violence done in the past, and more frequently of future and imaginary violence, while they themselves are the real offenders. You say that men committed robbery and murder in former times, and profess anxiety lest all men be robbed or murdered unless protected by your authority. This may or may not be true, but the fact that you allow thousands of men to perish in prisons by enforced labor, in fortresses, and in exile, that your military requisitions ruin millions of families and imperil, morally and physically, millions of men, this is not a supposititious but an actual violence, which, according to your own reasoning, should be resisted by violence. And therefore, by your own admission, the wicked ones, against whom one should use violence, are yourselves. Thus should the oppressed reply to their oppressors. And such are the language, the thoughts, and the actions of non-Christians. Wherever the oppressed are more wicked than the oppressor, they attack and overthrow them whenever they are able; or else—and this is more frequently the case—they enter the ranks of the oppressors and take part in their tyranny. Thus the dangers of which the defenders of State rights make a bugbear—that if authority were overthrown the wicked would prevail over the good—potentially exist at all times. The destruction of State violence, in fact, never can, for this very reason, lead to any real increase of violence on the part of the wicked over the good. If State violence disappeared, it is not unlikely that other acts of violence would be committed; but the sum of violence can never be increased simply because the power passes from the hands of one into those of another. State violence can never be abolished until all the wicked disappear, say the advocates of the existing order, by which they imply that there must always be violence, because there will always be wicked people. This could only prove true, supposing the oppressors to be really beneficent, and supposing the true deliverance of mankind from evil must be accomplished by violence.
- 57. Then, of course, violence could never cease. But as, on the contrary, violence never really overcomes evil, and since there is another way altogether to overcome it, the assertion that violence will never cease is untrue. Violence is diminishing, and clearly tending to disappear; though not, as is claimed by the defenders of the existing order, in consequence of the amelioration of those who live under an oppressive government (their condition really gets worse), but because the consciousness of mankind is becoming more clear. Hence even the wicked men who are in power are growing less and less wicked, and will at last become so good that they will be incapable of committing deeds of violence. The reason why humanity marches forward is not because the inferior men, having gained possession of power, reform their subjects by arbitrary methods, as is claimed both by Conservatives and Revolutionists, but is due above all to the fact that mankind in general is steadily, and with an ever increasing appreciation, adopting the Christian life-conception. There is a phenomenon observable in human life in a manner analogous to that of boiling. Those who profess the social life-conception are always ambitious to rule, and struggle to attain power. In this struggle the most gross and cruel, the least Christian elements of society, bubble up, as it were, and rise, by reason of their violence, into the ruling or upper classes of society. But then is fulfilled what Christ prophesied: Woe unto you that are rich! Woe unto you that are full! Woe unto you, when all men shall speak well of you! (Luke vi. 24-26). The men who have attained power, and glory, and riches, and who have realized all their cherished aims, live to discover that all is vanity, and gladly return to their former estate. Charles V., Ivan the Terrible, Alexander I., having realized the evils of power and its futility, renounced it because they recognized it as a calamity, having lost all pleasure in the deeds of violence which they formerly enjoyed. But it is not alone kings like Charles V. and Alexander I. who arrive at this disgust of power, but every man who has attained the object of his ambition. Not only the statesman, the general, the millionaire, the merchant, but every official who has gained the position for
- 58. which he has longed this half score of years, every well-to-do peasant who has saved one or two hundred roubles, finds at last the same disillusion. Not only individuals, but entire nations, mankind as a whole, have passed through this experience. The attractions of power and all it brings—riches, honors, luxury— seem to men really worth struggling for only until they are won; for no sooner does a man hold them within his grasp than they manifest their own emptiness and gradually lose their charm, like clouds, lovely and picturesque in outline seen from afar, but no sooner is one enveloped in them than all their beauty vanishes. Men who have obtained riches and power, those who have struggled for them, but more particularly those who have inherited them, cease to be greedy for power or cruel in its acquisition. Having learned by experience, sometimes in one generation, sometimes in several, how utterly worthless are the fruits of violence, men abandon those vices acquired by the passion for riches and power, and growing more humane, they lose their positions, being crowded out by others who are less Christian and more wicked; whereupon they fall back into a stratum, which, though lower in the social scale, is higher in that of morality, thus increasing the mean level of Christian consciousness. But straightway, the worse, the rougher, and less Christian elements rise to the surface, and being subject to the same experience as their predecessors, after one or two generations these men, too, recognize the hollowness of violent ambitions, and, being penetrated with the spirit of Christianity, fall back into the ranks of the oppressed. These are in turn replaced by new oppressors, less despotic than the former, but rougher than those whom they oppress. So that although the authority is to all outward seeming unchanged, yet the number of those who have been driven by the exigencies of life to adopt the Christian life-conception increases with every change of rulers. They may be more harsh, more cruel, and
- 59. less Christian than their subjects; but always men less and less violent replace their predecessors in authority. Violence chooses its instruments from among the worst elements of society; men who gradually become leavened, and, softened and changed for the better, are returned into society. Such is the process by means of which Christianity takes fuller possession of men day by day. Christianity enters into the consciousness of men in spite of the violence of power, and even owing to that violence. The argument of the defenders of the State, that if power were abolished the wicked would tyrannize over the good, not only fails to prove that the domination of the wicked is a new thing to be dreaded,—as it exists already,—but proves, on the contrary, that the tyranny of the State, which allows the wicked to govern the good, is itself the real evil which we ought to eradicate, and which is constantly decreasing by the very nature of things. But if State violence is not to cease until the rulers have become so far Christianized that they will renounce it of their own accord and no others will be found to take their places,—if these things are coming to pass, say the defenders of the existing order, when is it to happen? If 1800 years have passed, and still so many long to rule, it is wholly improbable that we shall soon behold this change, if it ever takes place at all. Even though there may be at present, as there always have been, certain individuals who would not rule if they could, who do not choose to benefit themselves in that way, still the number of those who do prefer to rule rather than to be ruled is so great that it is difficult to imagine a time when the number will be exhausted. In order to accomplish the conversion of all men, to induce each one to exchange the pagan for the Christian life-conception, voluntarily resigning riches and power, there being none left to profit by these, it would be necessary that not only all the rude, half- barbarous people, unfitted either to accept Christianity or follow its
- 60. precepts, who are always to be found in every Christian community, should become Christians, but that all savage and non-Christian nations, which are still numerous, should also become Christian. Therefore were one to admit that the Christianizing process may at some future time embrace all humanity, we must still take into consideration the degree of progress that has been made in 1800 years, and realize that this can only happen after many centuries. Hence we need not for the present trouble ourselves about the overthrow of authority; all we have to do is to look to it that it is in the best hands. Thus reply the partizans of the existing system. And this reasoning would be perfectly consistent, provided that the transition of men from one life-conception to another were only to be effected by the process of individual conversion; that is to say, that each man, through his personal experience, should realize the vanity of power, and apprehend Christian truth. This process is constantly going on, and in that way, one by one, men are converted to Christianity. But men do not become converted to Christianity merely in this way; there is an exterior influence brought to bear which accelerates the process. The progression of mankind from one system of life to another is accomplished not only gradually, as the sand glides through the hour-glass, grain by grain, until all has run out, but rather as water which enters an immersed vessel, at first slowly, at one side, then, borne down by its weight, suddenly plunges, and at once fills completely. And this is what happens in human communities during a change in their life-conception, which is equivalent to the change from one organization to another. It is only at first that men by degrees, one by one, accept the new truth and obey its dictates; but after it has been to a certain extent disseminated, it is accepted, not through intuition, and not by degrees, but generally and at once, and almost involuntarily.
- 61. And therefore the argument of the advocates of the present system, that but a minority have embraced Christianity during the last 1800 years, and that another 1800 years must pass away before the rest of mankind will accept it, is erroneous. For one must take into consideration another mode, in addition to the intuitive of assimilating new truth, and of making the transition from one mode of life to another. This other mode is this: men assimilate a truth not alone because they may have come to realize it through prophetic insight or through individual experience, but the truth having been spread abroad, those who dwell on a lower plane of intelligence accept it at once, because of their confidence in those who have received it and incorporated it in their lives. Every new truth that changes the manner of life and causes humanity to move onward is at first accepted by a very limited number, who grasp it by knowledge of it. The rest of mankind, accepting on faith the former truth upon which the existing system has been founded, is always opposed to the spread of the new truth. But as, in the first place, mankind is not stationary, but is ever progressing, growing more and more familiar with truth and approaching nearer to it in everyday life: and secondly, as all men progress according to their opportunities, age, education, nationality, beginning with those who are more, and ending with those who are less, capable of receiving new truth—the men nearest those who have perceived the truth intuitively pass, one by one, and with gradually diminishing intervals, over to the side of the new truth. So, as the number of men who acknowledge it increases, the truth itself becomes more clearly manifested. The feeling of confidence in the new truth increases in proportion to the numbers who have accepted it. For, owing to the growing intelligibility of the truth itself, it becomes easier for men to grasp it, especially for those lower intellectually, until finally the greater number readily adopt it, and help to found a new régime. The men who go over to the new truth, once it has gained a certain hold, go over en masse, of one accord, much as ballast is rapidly put
- 62. into a ship to maintain its equilibrium. If not ballasted, the vessel would not be sufficiently immersed, and would change its position every moment. This ballast, which at first may seem superfluous and a hindrance to the progress of the ship, is indispensable to its equipoise and motion. Thus it is with the masses when, under the influence of some new idea that has won social approval, they abandon one system to adopt another, not singly, but in a body. It is the inertia of this mass which impedes the rapid and frequent transition from one system of life, not ratified by wisdom, to another; and which for a long time arrests the progress of every truth destined to become a part of human consciousness. It is erroneous, then, to argue that because only a small percentage of the human race has in these eighteen centuries adopted the Christian doctrine, that many, many times eighteen centuries must elapse before the whole world will accept it,—a period of time so remote that we who are now living can have no interest in it. It is unfair, because those men who stand on a lower plane of development, whom the partizans of the existing order represent as hindrances to the realization of the Christian system of life, are those men who always go over in a body to a truth accepted by those above them. And therefore that change in the life of mankind, when the powerful will give up their power without finding any to assume it in their stead, will come to pass when the Christian life-conception, rendered familiar, conquers, not merely men one by one, but masses at a time. But even if it were true, the advocates of the existing order may say, that public opinion has the power to convert the inert non- Christian mass of men, as well as the corrupt and gross who are to be found in every Christian community, how shall we know that a Christian mode of life is born, and that State violence will be rendered useless?
- 63. After renouncing the despotism by which the existing order has been maintained, in order to trust to the vague and indefinite force of public opinion, we risk permitting those savages, those existing among us, as well as those outside, to commit robbery, murder, and other outrages upon Christians. If even with the help of authority we have a hard struggle against the anti-Christian elements ever ready to overpower us, and destroy all the progress made by civilization, how then could public opinion prove an efficient substitute for the use of force, and avail for our protection? To rely upon public opinion alone would be as foolhardy as to let loose all the wild beasts of a menagerie, because they seem inoffensive when in their cages and held in awe by red-hot irons. Those men entrusted with authority, or born to rule over others by the divine will of God, have no right to imperil all the results of civilization, simply to make an experiment, and learn whether public opinion can or cannot be substituted for the safeguard of authority. Alphonse Karr, a French writer, forgotten to-day, once said, in trying to prove the impossibility of abolishing the death penalty: Que Messieurs les assassins commencent par nous donner l'exemple. And I have often heard this witticism quoted by persons who really believed they were using a convincing and intellectual argument against the suppression of the penalty of death. Nevertheless, there could be no better argument against the violence of government. Let the assassins begin by showing us an example, say the defenders of government authority. The assassins say the same, but with more justice. They say: Let those who have set themselves up as teachers and guides show us an example by the suppression of legal assassination, and we will imitate it. And this they say, not by way of a jest, but in all seriousness, for such is in reality the situation. We cannot cease to use violence while we are surrounded by those who commit violence.
- 64. There is no more insuperable barrier at the present time to the progress of humanity, and to the establishment of a system that shall be in harmony with its present conception of life, than this erroneous argument. Those holding positions of authority are fully convinced that men are to be influenced and controlled by force alone, and therefore to preserve the existing system they do not hesitate to employ it. And yet this very system is supported, not by violence, but by public opinion, the action of which is compromised by violence. The action of violence actually weakens and destroys that which it wishes to support. At best, violence, if not employed as a vehicle for the ambition of those in high places, condemns in the inflexible form a law which public opinion has most probably long ago repudiated and condemned; but there is this difference, that while public opinion rejects and condemns all acts that are opposed to the moral law, the law supported by force repudiates and condemns only a certain limited number of acts, seeming thus to justify all acts of a like order which have not been included in its formula. From the time of Moses public opinion has regarded covetousness, lust, and cruelty as crimes, and condemned them as such. It condemns and repudiates every form that covetousness may assume, not only the acquisition of another man's property by violence, fraud, or cunning, but the cruel abuse of wealth as well. It condemns all kinds of lust, let it be impudicity with a mistress, a slave, a divorced wife, or with one's wife; it condemns all cruelty,— blows, bad usage, murder,—all cruelty, not only toward human beings, but toward animals. Whereas, the law, based upon violence, attacks only certain forms of covetousness, such as theft and fraud, and certain forms of lust and cruelty, such as conjugal infidelity, assault, and murder; and thus it seems to condone those manifestations of covetousness, lust, and cruelty which do not fall within its narrow limits.
- 65. But violence not only demoralizes public opinion, it excites in the minds of men a pernicious conviction that they move onward, not through the impulsion of a spiritual power, which would help them to comprehend and realize the truth by bringing them nearer to that moral force which is the source of every progressive movement of mankind,—but, by means of violence,—by the very factor that not only impedes our progress toward truth, but withdraws us from it. This is a fatal error, inasmuch as it inspires in man a contempt for the fundamental principle of his life,—spiritual activity,—and leads him to transfer all his strength and energy to the practice of external violence. It is as though men would try to put a locomotive in motion by turning its wheels with their hands, not knowing that the expansion of steam was the real motive-power, and that the action of the wheels was but the effect, and not the cause. If by their hands and their levers they move the wheels, it is but the semblance of motion, and, if anything, injures the wheels and makes them useless. The same mistake is made by those who expect to move the world by violence. Men affirm that the Christian life cannot be established save by violence, because there are still uncivilized nations outside of the Christian world, in Africa and Asia (some regard even the Chinese as a menace of our civilization), and because, according to the new theory of heredity, there exist in society congenital criminals, savage and irredeemably vicious. But the savages whom we find in our own community, as well as those beyond its pale, with whom we threaten ourselves and others, have never yielded to violence, and are not yielding to it now. One people never conquered another by violence alone. If the victors stood on a lower plane of civilization than the conquered, they always adopted the habits and customs of the latter, never attempting to force their own methods of life upon them. It is by the influence of public opinion, not by violence, that nations are reduced to submission.
- 66. When a people have accepted a new religion, have become Christians, or turned Mohammedans, it has come to pass, not because it was made obligatory by those in power (violence often produced quite the opposite result), but because they were influenced by public opinion. Nations constrained by violence to accept the religion of the conqueror have never really done so. The same may be said in regard to the savage elements found in all communities: neither severity nor clemency in the matter of punishments, nor modifications in the prison system, nor augmenting of the police force, have either diminished or increased the aggregate of crimes, which will only decrease through an evolution in our manner of life. No severities have ever succeeded in suppressing the vendetta, or the custom of dueling in certain countries. However many of his fellows may be put to death for thieving, the Tcherkess continues to steal out of vainglory. No girl will marry a Tcherkess who has not proved his daring by stealing a horse, or at least a sheep. When men no longer fight duels, and the Tcherkess cease to steal, it will not be from fear of punishment (the danger of capital punishment adds to the prestige of daring), but because public manners will have undergone a change. The same may be said of all other crimes. Violence can never suppress that which is countenanced by general custom. If public opinion would but frown upon violence, it would destroy all its power. What would happen if violence were not employed against hostile nations and the criminal element in society we do not know. But that the use of violence subdues neither we do know through long experience. And how can we expect to subdue by violence nations whose education, traditions, and even religious training all tend to glorify resistence to the conqueror, and love of liberty as the loftiest of virtues? And how is it possible to extirpate crime by violence in the midst of communities where the same act, regarded by the government as criminal, is transformed into an heroic exploit by public opinion?
- 67. Nations and races may be destroyed by violence—it has been done. They cannot be subdued. The power transcending all others which has influenced individuals and nations since time began, that power which is the convergence of the invisible, intangible, spiritual forces of all humanity, is public opinion. Violence serves but to enervate this influence, disintegrating it, and substituting for it one not only useless, but pernicious to the welfare of humanity. In order to win over all those outside the Christian fold, all the Zulus, the Manchurians, the Chinese, whom many consider uncivilized, and the uncivilized among ourselves, there is only one way. This is by the diffusion of a Christian mode of thought, which is only to be accomplished by a Christian life, Christian deeds, a Christian example. But instead of employing this one way of winning those who have remained outside the fold of Christianity, men of our epoch have done just the opposite. In order to convert uncivilized nations who do us no harm, whom we have no motive for oppressing, we ought, above all, to leave them in peace, and act upon them only by our showing them an example of the Christian virtues of patience, meekness, temperance, purity, and brotherly love. Instead of this we begin by seizing their territory, and establishing among them new marts for our commerce, with the sole view of furthering our own interests—we, in fact, rob them; we sell them wine, tobacco, and opium, and thereby demoralize them; we establish our own customs among them, we teach them violence and all its lessons; we teach them the animal law of strife, that lowest depth of human degradation, and do all that we can to conceal the Christian virtues we possess. Then, having sent them a score of missionaries, who gabble an absurd clerical jargon, we quote the results of our attempt to convert the heathen as an indubitable proof that the truths of Christianity are not adaptable to everyday life.
- 68. And as for those whom we call criminals, who live in our midst, all that has just been said applies equally to them. There is only one way to convert them, and that is by means of a public opinion founded on true Christianity, accompanied by the example of a sincere Christian life. And by way of preaching this Christian gospel and confirming it by Christian example, we imprison, we execute, guillotine, hang; we encourage the masses in idolatrous religions calculated to stultify them; the government authorizes the sale of brain-destroying poisons—wine, tobacco, opium; prostitution is legalized; we bestow land upon those who need it not; surrounded by misery, we display in our entertainments an unbridled extravagance; we render impossible in such ways any semblance of a Christian life, and do our best to destroy Christian ideas already established; and then, after doing all we can to demoralize men, we take and confine them like wild beasts in places from which they cannot escape, and where they will become more brutal than ever; or we murder the men we have demoralized, and then use them as an example to illustrate and prove our argument that people are only to be controlled by violence. Even so does the ignorant physician act, who, having placed his patient in the most unsanitary conditions, or having administered to him poisonous drugs, afterward contends that his patient has succumbed to the disease, when had he been left to himself he would have recovered long ago. Violence, which men regard as an instrument for the support of Christian life, on the contrary, prevents the social system from reaching its full and perfect development. The social system is such as it is, not because of violence, but in spite of it. Therefore the defenders of the existing social system are self- deceived when they say that, since violence barely holds the evil and un-Christian elements of society in awe, its subversion, and the substitution of the moral influence of public opinion, would leave us helpless in face of them. They are wrong, because violence does not protect mankind; but it deprives men of the only possible chance of
- 69. an effectual defense by the establishment and propagation of the Christian principle of life. But how can one discard the visible and tangible protection of the policeman with his baton, and trust to invisible, intangible public opinion? And, moreover, is not its very existence problematical? We are all familiar with the actual state of things; whether it be good or bad we know its faults, and are accustomed to them; we know how to conduct ourselves, how to act in the present conditions; but what will happen when we renounce the present organization, and confide ourselves to something invisible, intangible, and utterly unfamiliar? Men dread the uncertainty into which they would plunge if they were to renounce the familiar order of things. Certainly were our situation an assured and stable one, it would be well to dread the uncertainties of change. But so far from enjoying an assured position, we know that we are on the verge of a catastrophe. If we are to give way to fear, then let it be before something that is really fearful, and not before something that we imagine may be so. In fearing to make an effort to escape from conditions that are fatal to us, only because the future is obscure and unknown, we are like the passengers of a sinking ship who crowd into the cabin and refuse to leave it, because they have not the courage to enter the boat that would carry them to the shore; or like sheep who, in fear of the fire that has broken out in the farmyard, huddle together in a corner and will not go out through the open gate. How can we, who stand on the threshold of a shocking and devastating social war, before which, as those who are preparing for it tell us, the horrors of 1793 will pale, talk seriously about the danger threatened by the natives of Dahomey, the Zulus, and others who live far away, and who have no intention of attacking us; or about the few thousands of malefactors, thieves, and murderers— men whom we have helped to demoralize, and whose numbers are not decreased by all our courts, prisons, and executions?
- 70. Moreover, this anxiety lest the visible protection of the police be overthrown, is chiefly confined to the inhabitants of cities—that is, to those who live under abnormal and artificial conditions. Those who live normally in the midst of nature, dealing with its forces, require no such protection; they realize how little avails violence to protect us from the real danger that surrounds us. There is something morbid in this fear, which arises chiefly from the false conditions in which most of us have grown up and continue to live. A doctor to the insane related how, one day in summer, when he was about to leave the asylum, the patients accompanied him as far as the gate that led into the street. Come with me into town! he proposed to them. The patients agreed, and a little band followed him. But the farther they went through the streets where they met their sane fellow-men moving freely to and fro, the more timid they grew, and pressed more closely around the doctor. At last they begged to be taken back to the asylum, to their old but accustomed mode of insane life, to their keepers and their rough ways, to strait jackets and solitary confinement. And thus it is with those whom Christianity is waiting to set free, to whom it offers the untrammeled rational life of the future, the coming century; they huddle together and cling to their insane customs, to their factories, courts, and prisons, their executioners, and their warfare. They ask: What security will there be for us when the existing order has been swept away? What kind of laws are to take the place of those under which we are now living? Not until we know exactly how our life is to be ordered will we take a single step toward making a change. It is as if a discoverer were to insist upon a detailed description of the region he is about to explore. If the individual man, while passing from one period of his life to another, could read the future and know just what his whole life were to be, he would have no reason for living. And so it is with the career of humanity. If,
- 71. upon entering a new period, a program detailing the incidents of its future existence were possible, humanity would stagnate. We cannot know the conditions of the new order of things, because we have to work them out for ourselves. The meaning of life is to search out that which is hidden, and then to conform our activity to our new knowledge. This is the life of the individual as it is the life of humanity.
- 72. CHAPTER XI CHRISTIAN PUBLIC OPINION ALREADY ARISES IN OUR SOCIETY, AND WILL INEVITABLY DESTROY THE SYSTEM OF VIOLENCE OF OUR LIFE. WHEN THIS WILL COME ABOUT The condition and organization of our society is shocking; it is upheld by public opinion, but can be abolished by it—Men's views in regard to violence have already changed; the number of men ready to serve the governments decreases, and functionaries of government themselves begin to be ashamed of their position, to the point of often not fulfilling their duties—These facts, signs of the birth of a public opinion, which, in becoming more and more general, will lead finally to the impossibility of finding men willing to serve governments—It becomes more and more clear that such positions are no longer needed—Men begin to realize the uselessness of all the institutions of violence; and if this is realized by a few men, it will later be understood by all—The time when the deliverance will be accomplished is unknown, but it depends on men themselves; it depends on how much each man is willing to live by the light that is within him. The position of the Christian nations, with their prisons, their gallows, their factories, their accumulations of capital, taxes, churches, taverns, and public brothels, their increasing armaments, and their millions of besotted men, ready, like dogs, to spring at a word from the master, would be shocking indeed if it were the result of violence; but such a state of things is, before all, the result of public opinion; and what has been established by public opinion not only may be, but will be, overthrown by it. Millions and millions of money, tens of millions of disciplined soldiers, marvelous weapons of destruction, an infinitely perfected
- 73. organization, legions of men charged to delude and hypnotize the people,—this is all under the control of men who believe that this organization is advantageous for them, who know that without it they would disappear, and who therefore devote all their energy to its maintenance. What an indomitable array of power it seems! And yet we have but to realize whither we are fatally tending, for men to become as much ashamed of acts of violence, and to profit by them, as they are ashamed now of dishonesty, theft, beggary, cowardice; and the whole complicated and apparently omnipotent system will die at once without any struggle. To accomplish this transformation it is not necessary that any new ideas should find their way into the human consciousness, but only that the mist which now veils the true significance of violence should lift, in order that the growing Christian public opinion and methods may conquer the methods of the pagan world. And this is gradually coming to pass. We do not observe it, as we do not observe the movement of things when we are turning, and everything around us is turning as well. It is true that the social organization seems for the most part as much under the influence of violence as it seemed a thousand years ago, and in respect of armaments and war seems even more; but the Christian view of life is already having its effect. The withered tree, to all appearance, stands as firmly as ever; it seems even firmer, because it has grown harder, but it is already rotten at the heart and preparing to fall. It is the same with the present mode of life based upon violence. The outward position of man appears the same. There are the same oppressors, the same oppressed, but the feeling of both classes in regard to their respective positions has undergone a change. The oppressors, that is, those who take part in the government, and those who are benefited by oppression, the wealthy classes, do not constitute, as formerly, the élite of society, nor does their condition suggest that ideal of human prosperity and greatness to which formerly all the oppressed aspired. Now, it often happens that the oppressors renounce of their own accord the advantages of their position, choosing the position of the oppressed,
- 74. and endeavor, by the simplicity of their mode of life, to resemble them. Not to speak of those offices and positions generally considered contemptible, such as that of the spy, the detective, the usurer, or the keeper of a tavern, a great many of the positions held by the oppressors, and formerly considered honorable, such as those of police officers, courtiers, judges, administrative functionaries, ecclesiastical or military, masters on a large scale, and bankers, are not only considered little enviable, but are already avoided by estimable men. Already there are men who choose to renounce such once envied positions, preferring others which, although less advantageous, are not associated with violence. It is not merely such as these who renounce their privileges; men influenced, not by religious motives, as was the case in former ages, but by growing public opinion, refuse to accept fortunes fallen to them by inheritance, because they believe that a man ought to possess only the fruits of his own labor. High-minded youths, not as yet depraved by life, when about to choose a career, prefer the professions of doctors, engineers, teachers, artists, writers, or even of farmers, who live by their daily toil, to the positions of judges, administrators, priests, soldiers in the pay of government; they decline even the position of living on their income. Most of the monuments at the present day are no longer erected in honor of statesmen or generals, still less of men of wealth, but to scientists, artists, and inventors, to men who not only had nothing in common with government or authority, but who frequently opposed it. It is to their memory that the arts are thus consecrated. The class of men who will govern, and of rich men, tends every day to grow less numerous, and so far as intellect, education, and especially morality, are concerned, rich men and men in power are not the most distinguished members of society, as was the case in olden times. In Russia and Turkey, as in France and America,
- 75. notwithstanding the frequent changes of officials, the greater number are often covetous and venal, and so little to be commended from the point of view of morality that they do not satisfy even the elementary exigencies of honesty demanded in government posts. Thus one hears often the ingenuous complaints of those in government that the best men among us, strangely enough as it seems to them, are always found among those opposed to them. It is as if one complained that it is not the nice, good people who become hangmen. Rich men of the present day, as a general thing, are mere vulgar amassers of wealth, for the most part having but little care beyond that of increasing their capital, and that most often by impure means; or are the degenerate inheritors, who, far from playing an important part in society, often incur general contempt. Many positions have lost their ancient importance. Kings and emperors now hardly direct at all; they seldom effect internal changes or modify external policy, leaving the decision of such questions to the departments of State, or to public opinion. Their function is reduced to being the representatives of state unity and power. But even this duty they begin to neglect. Most of them not only fail to maintain themselves in their former unapproachable majesty, but they grow more and more democratic, they prefer even to be bourgeois; they lay down thus their last distinction, destroying precisely what they are expected to maintain. The same may be said of the army. The high officers, instead of encouraging the roughness and cruelty of the soldiers, which befit their occupation, promote the diffusion of education among them, preach humanity, often sympathize with the socialistic ideas of the masses, and deny the utility of war. In the late conspiracies against the Russian government many of those concerned were military men. It often happens, as it did recently, that the troops, when called upon to establish order, refuse to fire on the people. The barrack code of ideas is frankly deprecated by military men themselves, who often enough make it the subject of derision.
- 76. The same may be said of judges and lawyers. Judges, whose duty it is to judge and condemn criminals, conduct their trials in such a fashion as to prove them innocent; thus the Russian government, when it desires the condemnation of those it wishes to punish, never confides them to the ordinary tribunals; it tries them by court- martial, which is but a parody of justice. The same may be said of lawyers, who often refuse to accuse, and, twisting round the law, defend those they should accuse. Learned jurists, whose duty it is to justify the violence of authority, deny more and more frequently the right of punishment, and in its place introduce theories of irresponsibility, often prescribing, not punishment, but medical treatment for so-called criminals. Jailers and turnkeys in convict prisons often become the protectors of those it is a part of their business to torture. Policemen and detectives are constantly saving those they ought to arrest. Ecclesiastics preach tolerance; they often deny the right of violence, and the more educated among them attempt in their sermons to avoid the deception which constitutes all the meaning of their position, and which they are expected to preach. Executioners refuse to perform their duty; the result is that often in Russia death- warrants cannot be carried out for lack of executioners, for, notwithstanding all the advantages of the position, the candidates, who are chosen from convicts, diminish in number every year. Governors, commissioners, and tax-collectors, pitying the people, often try to find pretexts for remitting the taxes. Rich men no longer dare to use their wealth for themselves alone, but sacrifice a part of it to social charities. Landowners establish hospitals and schools on their estates, and some even renounce their estates and bestow them on the cultivators of the soil, or establish agricultural colonies upon them. Manufacturers and mill-owners found schools, hospitals, and savings-banks, institute pensions, and build houses for the workmen; some start associations of which the profits are equally divided among all. Capitalists expend a portion of their wealth on educational, artistic, and philanthropic institutions for the public
- 77. benefit. Many men who are unwilling to part with their riches during their lifetime bequeath them to public institutions. These facts might be deemed the result of chance were it not that they all originate from one source, as, when certain trees begin to bud in the spring of the year, we might believe it accidental, only we know the cause; and that if on some trees the buds begin to swell, we know that the same thing will happen to all of them. Even so is it in regard to Christian public opinion and its manifestations. If this public opinion already influences some of the more sensitive men, and makes each one in his own sphere decline the advantages obtained by violence or its use, it will continue to influence men more and more, until it brings about a change in their mode of life and reconciles it with that Christian consciousness already possessed by the most advanced. And if there are already rulers who do not venture on any undertaking on their own responsibility, and who try to be like ordinary men rather than monarchs, who declare themselves ready to give up their prerogatives and become the first citizens of their country, and soldiers who, realizing all the sin and evil of war, do not wish to kill either foreigners or their fellow-countrymen, judges and lawyers who do not wish to accuse and condemn criminals, priests who evade preaching lies, tax-gatherers who endeavor to fulfil as gently as possible what they are called upon to do, and rich men who give up their wealth, then surely it will ultimately come to pass that other rulers, soldiers, priests, and rich men will follow their example. And when there are no more men ready to occupy positions supported by violence, the positions themselves will cease to exist. But this is not the only way by which public opinion leads toward the abolition of the existing system, and the substitution of a new one. As the positions supported by violence become by degrees less and less attractive, and there are fewer and fewer applicants to fill them, their uselessness becomes more and more apparent.
- 78. We have to-day the same rulers and governments, the same armies, courts of law, tax-gatherers, priests, wealthy landowners, manufacturers, and capitalists as formerly, but their relative positions are changed. The same rulers go about to their various interviews, they have the same meetings, hunts, festivities, balls, and uniforms; the same diplomatists have the same conversations about alliances and armies; the same parliaments, in which Eastern and African questions are discussed, and questions in regard to alliances, ruptures, Home Rule, the eight-hour day. Changes of ministry take place just as of old, accompanied by the same speeches and incidents. But to those who know how an article in a newspaper changes perhaps the position of affairs more than dozens of royal interviews and parliamentary sessions, it becomes more and more evident that it is not these meetings, interviews, and parliamentary discussions that control affairs, but something independent of all this, something which has no local habitation. The same generals, officers, soldiers, cannon, fortresses, parades, and evolutions. But one year elapses, ten, twenty years elapse, and there is no war. And troops are less and less to be relied on to suppress insurrection, and it becomes more and more evident that generals, officers, and soldiers are only figure-heads in triumphal processions, the plaything of a sovereign, a sort of unwieldy and expensive corps-de-ballet. The same lawyers and judges, and the same sessions, but it becomes more and more evident that as civil courts make decisions in a great variety of causes without anxiety about purely legal justice, and that criminal courts are useless, because the punishment does not produce the desired result, therefore these institutions have no other object than the maintenance of men incapable of doing other things more useful. The same priests, bishops, churches, and synods, but it becomes more and more evident to all that these men themselves have long since ceased to believe what they preach, and are therefore unable
- 79. to persuade any one of the necessity of believing what they no longer believe themselves. The same tax-gatherers, but more and more incapable of extorting money from the people by force, and it becomes more and more evident that, without such collectors, it would be possible to obtain by voluntary contribution all that is required for social needs. The same rich men, and yet it becomes more and more evident that they can be useful only when they cease to be personal administrators of their possessions, and surrender to society their wealth in whole or part. When this becomes as plain to all men as it now is to a few, the question will naturally arise: Why should we feed and support all those emperors, kings, presidents, members of departments, and ministers, if all their interviews and conversations amount to nothing? Would it not be better, as some wit expressed it, to set up an india-rubber queen? And of what use to us are armies, with their generals, their musicians, their horses, and drums? Of what use are they when there is no war, when no one wishes to conquer anybody else? And even if there were a war, other nations would prevent us from reaping its advantages; while upon their compatriots the troops would refuse to fire. And what is the use of judges and attorneys whose decisions in civil cases are not according to the law, and who, in criminal ones, are aware that punishments are of no avail? And of what use are tax-gatherers who are reluctant to collect the taxes, when all that is needed could be contributed without their assistance? And where is the use of a clergy which has long ceased to believe what it preaches? And of what use is capital in the hands of private individuals when it can be beneficial only when it becomes public property? Having once
- 80. asked all these questions, men cannot but arrive at the conclusion that institutions which have lost their usefulness should no longer be supported. And furthermore, men who themselves occupy positions of privilege come to see the necessity of abandoning them. One day, in Moscow, I was present at a religious discussion which is usually held during St. Thomas's week, near the church in the Okhotny Ryad. A group of perhaps twenty men had gathered on the pavement, and a serious discussion concerning religion was in progress. Meanwhile, in the nobles' club near at hand, a concert was taking place, and a police-officer, having noticed the group of people gathered near the church, sent a mounted policeman to order them to disperse,—not that the police-officer cared in the least whether the group stayed where it was or dispersed. The twenty men who had gathered inconvenienced no one, but the officer had been on duty all the morning and felt obliged to do something. The young policeman, a smart-looking fellow, with his right arm akimbo and a clanking sword, rode up to us, calling out in an imperative tone: Disperse, you fellows! What business have you to gather there? Every one turned to look at him, while one of the speakers, a modest-looking man in a peasant's coat, replied calmly and pleasantly: We are talking about business, and there is no reason why we should disperse; it might be better for you, my young friend, if you were to jump off from your horse and to listen to us. Very likely it would do you good; and turning away he continued the conversation. The policeman turned his horse without a word and rode away. Such scenes as this must be of frequent occurrence in countries where violence is employed. The officer was bored; he had nothing to do, and the poor fellow was placed in a position where he felt in duty bound to give orders. He was deprived of a rational human existence; he could do nothing but look on and give orders, give orders and look on, although both were works of supererogation. It will not be long before all those unfortunate rulers, ministers,
- 81. members of parliaments, governors, generals, officers, bishops, priests, and even rich men, will find themselves—indeed they have already done so—in precisely the same position. Their sole occupation consists in issuing orders; they send out their subordinates, like the officer who sent the policeman to interfere with the people; and as the people with whom they interfere ask not to be interfered with, this seems to their official intelligence only to prove that they are very necessary. But the time will surely come when it will be perfectly evident to every one that they are not only useless, but an actual impediment, and those whose course they obstruct will say gently and pleasantly, like the man in the peasant's coat: We beg that you will let us alone. Then the subordinates as well as their instructors will find themselves compelled to take the good advice that is offered them, cease to prance about among men with their arms akimbo, and having discarded their glittering livery, listen to what is said among men, and unite with them to help to promote the serious work of the world. Sooner or later the time will surely come when all the present institutions supported by violence will cease to be; their too evident uselessness, absurdity, and even unseemliness, will finally destroy them. There must come a time when the same thing that happened to the king in Andersen's fairy tale, The King's New Clothes, will happen to men occupying positions created by violence. The tale tells of a king who cared enormously for new clothes, and to whom one day came two tailors who agreed to make him a suit woven from a wonderful stuff. The king engaged them and they set to work, saying that the stuff possessed the remarkable quality of becoming invisible to any one unfit for the office he holds. The courtiers came to inspect the work of the tailors, but could see nothing, because these men were drawing their needles through empty space. However, remembering the consequences, they all pretended to see the cloth and to be very much pleased with it.
- 82. Even the king himself praised it. The hour appointed for the procession when he was to walk wearing his new garment arrived. The king took off his clothes and put on the new ones—that is, he remained naked all the while, and thus he went in procession. But remembering the consequences, no one had the courage to say that he was not dressed, until a little child, catching sight of the naked king, innocently exclaimed, But he has nothing on! Whereupon all the others who had known this before, but had not acknowledged it, could no longer conceal the fact. Thus will it be with those who, through inertia, continue to fill offices that have long ceased to be of any consequence, until some chance observer, who happens not to be engaged, as the Russian proverb has it, in washing one hand with the other, will ingenuously exclaim, It is a long time since these men were good for anything! The position of the Christian world, with its fortresses, cannon, dynamite, guns, torpedoes, prisons, gallows, churches, factories, custom-houses, and palaces is monstrous. But neither fortresses nor cannon nor guns by themselves can make war, nor can the prisons lock their gates, nor the gallows hang, nor the churches themselves lead men astray, nor the custom-houses claim their dues, nor palaces and factories build and support themselves; all these operations are performed by men. And when men understand that they need not make them, then these things will cease to be. And already men are beginning to understand this. If not yet understood by all, it is already understood by those whom the rest of the world eventually follows. And it is impossible to cease to understand what once has been understood, and the masses not only can, but inevitably must, follow where those who have understood have already led the way. Hence the prophecy: that a time will come when all men will hearken unto the word of God, will forget the arts of war, will melt their swords into plowshares and their lances into reaping-hooks;— which, being translated, means when all the prisons, the fortresses, the barracks, the palaces, and the churches will remain empty, the
- 83. gallows and the cannon will be useless. This is no longer a mere Utopia, but a new and definite system of life, toward which mankind is progressing with ever increasing rapidity. But when will it come? Eighteen hundred years ago Christ, in answer to this question, replied that the end of the present world—that is, of the pagan system—would come when the miseries of man had increased to their utmost limit; and when, at the same time, the good news of the Kingdom of Heaven—that is, of the possibility of a new system, one not founded upon violence—should be proclaimed throughout the earth.[20] But of that day and hour knoweth no man, no, not the angels of heaven, but my Father only,[21] said Christ. Watch therefore: for ye know not what hour your Lord doth come. When will the hour arrive? Christ said that we cannot know. And for that very reason we should hold ourselves in readiness to meet it, as the goodman should watch his house against thieves, or like the virgins who await with their lamps the coming of the bridegroom; and, moreover, we should work with all our might to hasten the coming of that hour, as the servants should use the talents they have received that they may increase.[22] And there can be no other answer. The day and the hour of the advent of the Kingdom of God men cannot know, since the coming of that hour depends only on men themselves. The reply is like that of the wise man who, when the traveler asked him how far he was from the city, answered, Go on! How can we know if it is still far to the goal toward which humanity is aiming, when we do not know how it will move toward it; that it depends on humanity whether it moves steadily onward or pauses, whether it accelerates or retards its pace. All that we can know is what we who form humanity should or should not do in order to bring about this Kingdom of God. And that
- 84. we all know; for each one has but to begin to do his duty, each one has but to live according to the light that is within him, to bring about the immediate advent of the promised Kingdom of God, for which the heart of every man yearns.
- 85. CHAPTER XII CONCLUSION REPENT, FOR THE KINGDOM OF HEAVEN IS AT HAND! 1 Encounter with a train carrying soldiers to establish order among famine- stricken peasants—The cause of the disorder—How the mandates of the higher authorities are carried out in case of peasants' resistance—The affair at Orel as an example of violence and murder committed for the purpose of asserting the rights of the rich—All the advantages of the rich are founded on like acts of violence. 2 The Tula train and the behavior of the persons composing it—How men can behave as these do—The reasons are neither ignorance, nor cruelty, nor cowardice, nor lack of comprehension or of moral sense—They do these things because they think them necessary to maintain the existing system, to support which they believe to be every man's duty—On what the belief of the necessity and immutability of the existing order of things is founded—For the upper classes it is based on the advantages it affords them—But what compels men of the lower classes to believe in the immutability of this system, when they derive no advantage from it, and maintain it with acts contrary to their conscience?—The reason lies in the deceit practised by the upper classes upon the lower in regard to the necessity of the existing order, and the legitimacy of acts of violence for its maintenance—General deception—Special deception—The conscription. 3 How men reconcile the legitimacy of murder with the precepts of morality, and how they admit the existence in their midst of a military organization for purposes of violence which incessantly threatens the
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