![Preface
Interactions between the theory of partial differential equations of elliptic and
parabolic types and the theory of stochastic processes are beneficial for, both, prob-
ability theory and analysis. At the beginning, mostly analytic results were used by
probabilists. More recently, the analysts (and physicists) took inspiration from the
probabilistic approach. Of course, the development of analysis, in general, and of
theory of partial differential equations, in particular, was motivated to a great ex-
tent by the problems in physics. A difference between physics and probability is
that the latter provides not only an intuition but also rigorous mathematical tools
for proving theorems.
The subject of this book is connections between linear and semilinear differ-
ential equations and the corresponding Markov processes called diffusions and su-
perdiffusions. A diffusion is a model of a random motion of a single particle. It is
characterized by a second order elliptic differential operator L. A special case is the
Brownian motion corresponding to the Laplacian ∆. A superdiffusion describes a
random evolution of a cloud of particles. It is closely related to equations involving
an operator Lu − ψ(u). Here ψ belongs to a class of functions which contains, in
particular ψ(u) = uα
with α > 1. Fundamental contributions to the analytic theory
of equations
(0.1) Lu = ψ(u)
and
(0.2) u̇ + Lu = ψ(u)
were made by Keller, Osserman, Brezis and Strauss, Loewner and Nirenberg, Brezis
and Véron, Baras and Pierre, Marcus and Véron.
A relation between the equation (0.1) and superdiffusions was established, first,
by S. Watanabe. Dawson and Perkins obtained deep results on the path behavior
of the super-Brownian motion. For applying a superdiffusion to partial differential
equations it is insufficient to consider the mass distribution of a random cloud at
fixed times t. A model of a superdiffusion as a system of exit measures from time-
space open sets was developed in [Dyn91c], [Dyn92], [Dyn93]. In particular,
a branching property and a Markov property of such system were established and
used to investigate boundary value problems for semilinear equations. In the present
book we deduce the entire theory of superdiffusion from these properties.
We use a combination of probabilistic and analytic tools to investigate positive
solutions of equations (0.1) and (0.2). In particular, we study removable singulari-
ties of such solutions and a characterization of a solution by its trace on the bound-
ary. These problems were investigated recently by a number of authors. Marcus
and Véron used purely analytic methods. Le Gall, Dynkin and Kuznetsov combined
7](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-11-320.jpg)
![8 PREFACE
probabilistic and analytic approach. Le Gall invented a new powerful probabilistic
tool — a path-valued Markov process called the Brownian snake. In his pioneer-
ing work he used this tool to describe all solutions of the equation ∆u = u2
in a
bounded smooth planar domain.
Most of the book is devoted to a systematic presentation (in a more general
setting, with simplified proofs) of the results obtained since 1988 in a series of papers
of Dynkin and Dynkin and Kuznetsov. Many results obtained originally by using
superdiffusions are extended in the book to more general equations by applying a
combination of diffusions with purely analytic methods. Almost all chapters involve
a mixture of probability and analysis. Exceptions are Chapters 7 and 9 where the
probability prevails and Chapter 13 where it is absent. Independently of the rest of
the book, Chapter 7 can serve as an introduction to the Martin boundary theory for
diffusions based on Hunt’s ideas. A contribution to the theory of Markov processes
is also a new form of the strong Markov property in a time inhomogeneous setting.
The theory of parabolic partial differential equations has a lot of similarities
with the theory of elliptic equations. Many results on elliptic equations can be easily
deduced from the results on parabolic equations. On the other hand, the analytic
technique needed in the parabolic setting is more complicated and the most results
are easier to describe in the elliptic case.
We consider a parabolic setting in Part 1 of the book. This is necessary for
constructing our principal probabilistic model — branching exit Markov systems.
Superprocesses (including superdiffusions) are treated as a special case of such sys-
tems. We discuss connections between linear parabolic differential equations and
diffusions and between semilinear parabolic equations and superdiffusions. (Diffu-
sions and superdiffusions in Part 1 are time inhomogeneous processes.)
In Part 2 we deal with elliptic differential equations and with time-homogeneous
diffusions and superdiffusions. We apply, when it is possible, the results of Part
1. The most of Part 2 is devoted to the characterization of positive solutions of
equation (0.1) by their traces on the boundary and to the study of the boundary
singularities of such solutions (both, from analytic and probabilistic points of view).
Parabolic counterparts of these results are less complete. Some references to them
can be found in bibliographical notes in which we describe the relation of the
material presented in each chapter to the literature on the subject.
Chapter 1 is an informal introduction where we present some of the basic ideas
and tools used in the rest of the book. We consider an elliptic setting and, to
simplify the presentation, we restrict ourselves to a particular case of the Laplacian
∆ (for L) and to the Brownian and super-Brownian motions instead of general
diffusions and superdiffusions.
In the concluding chapter, we give a brief description of some results not in-
cluded into the book. In particular, we describe briefly Le Gall’s approach to
superprocesses via random snakes (path-valued Markov processes). For a system-
atic presentation of this approach we refer to [Le 99a]. We do not touch some
other important recent directions in the theory of measure-valued processes: the
Fleming-Viot model, interactive measure-valued models... We refer on these sub-
jects to Lecture Notes of Dawson [Daw93] and Perkins [Per01]. A wide range of
topics is covered (mostly, in an expository form) in “An introduction to Superpro-
cesses” by Etheridge [Eth00].](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-12-320.jpg)
![CHAPTER 1
Introduction
1. Brownian and super-Brownian motions and differential equations
1.1. Brownian motion and Laplace equation. Let D be a bounded do-
main in Rd
with smooth boundary ∂D and let f be a continuous function on ∂D.
Then there exists a unique function u of class C2
such that
∆u = 0 in D,
u = f on ∂D.
(1.1)
It is called the solution of the Dirichlet problem for the Laplace equation in
D with the boundary value f. A probabilistic approach to this problem can be
traced to the classical work [CFL28] of Courant, Friedrichs and Lewy published in
1928. The authors replaced the Laplacian ∆ by its lattice approximation and they
represented the solution of the corresponding boundary value problem in terms of
the random walk on the lattice. Suppose that a particle starts from a site x in D
and moves in one step from a site x to any of 2d nearest neighbor sites with equal
probabilities. Let τ be its first exit time from D and ξτ be its location at time τ.
Then the solution of the Dirichlet problem on the lattice is given by the formula
(1.2) u(x) = Πxf(ξτ ) =
Z
f(ξτ(ω)(ω))Πx(dω),
where Πx is the probability distribution in the path space Ω corresponding to the
initial point x. The solution of the problem (1.1) can be obtained by the passage
to the limit as the lattice mesh and the duration of each step tend to 0 in a certain
relation.
In fact, this passage to the limit yields a measure Πx on the space of continuous
paths. The stochastic process ξ = (ξt, Πx) is called the Brownian motion and
formula (1.2) gives an explicit solution of the problem (1.1) in terms of the Brownian
motion ξ. This result is due to Kakutani [Kak44a], [Kak44b].
1.2. Semilinear equations. Partial differential equations involving a nonlin-
ear operator ∆u − ψ(u) appeared in meteorology (Emden, 1897), theory of atomic
spectra (Thomas-Fermi, 1920s) and astrophysics (Chandrasekhar, 1937). 1
Since the 1960s, geometers have been interested in these equations in connection
with the Yamabe problem: which two functions represent scalar curvature of two
Riemannian metrics related by a conformal mapping.
The equation
(1.3) ∆u = ψ(u)
1See the bibliography in [Vér96].
11](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-15-320.jpg)
![12 1. INTRODUCTION
was investigated under various conditions on the function ψ. All these conditions
hold for the family
(1.4) ψ(u) = uα
, α > 1.
For a wide class of ψ, the problem
∆u = ψ(u) in D,
u = f on ∂D,
(1.5)
has a unique solution under the same conditions on D and f as the classical problem
(1.1). However, analysts discovered a number of new phenomena related to this
equation. In 1957 Keller [Kel57a] and Osserman [Oss57] found that all positive
solutions of (1.3) are uniformly locally bounded. The most work was devoted to
the case of ψ given by (1.4). In 1974, Loewner and Nirenberg [LN74] proved that,
in an arbitrary domain D, there exists the maximal solution. This solution tends to
∞ at ∂D if D is bounded and ∂D is smooth. 2
In 1980 Brezis and Véron [BV80]
showed that the maximal solution in the punctured space Rd
{0} is trivial if
d ≥ κα =
2α
α − 1
and it is equal to
q | x |−2/(α−1)
with
q = [2(α − 1)−1
(κα − d)]1/(α−1)
if d < κα.
1.3. Super-Brownian approach to semilinear equations. A probabilistic
formula (1.2) for solving the problem (1.1) involves the value of f at a random point
ξτ on the boundary. The problem (1.5) can be approached by introducing, instead,
a random measure XD on ∂D and by taking the integral hf, XDi of f with respect
to XD. The probability law Pµ of XD depends on an initial measure µ and the role
similar to that of (1.2) is played by the formula
(1.6) u(x) = − logPxe−hf,XD i
.
Here Px stands for Pµ corresponding to the initial state µ = δx (unit mass con-
centrated at x). We call (XD, Pµ) the exit measure from D. Heuristically, we can
think of a random cloud for which XD is the mass distribution on an absorbing
barrier placed on ∂D.
We consider families of exit measures which we call branching exit Markov
(shortly, BEM) systems because their principal characteristics are a branching prop-
erty and a Markov property. The BEM system used in formula (1.6) is called the
super-Brownian motion. In the next section we explain how it can be obtained by
a passage to the limit from discrete BEM systems. Before that, we give, as the first
application of (1.6), an expression for a solution exploding on the boundary. Note
that, if XD 6= 0, then e−ch1,XDi
→ 0 as c → +∞ and, if XD = 0, then e−ch1,XDi
= 1
for all c. Therefore a solution tending to ∞ at ∂D can be expressed by the formula
(1.7) u(x) = − log Px{XD = 0}.
2They considered, in connection with a geometric problem, a special case α = d+2
d−2
.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-16-320.jpg)
![14 1. INTRODUCTION
the limit, we get an initial measure on D and an exit measure on ∂D which are
not discrete. We denote them again µ and XD. The branching property and the
Markov property are preserved under this passage to the limit and we get a BEM
system (XD, Pµ) where D is an arbitrary bounded open set and µ is an arbitrary
finite measure.
A function ψ obtained by a passage to the limit from ϕ belongs to a class Ψ0
which contains ψ(u) = uα
with 1 < α ≤ 2 but not with α > 2. The probability
distribution of the random measure (XD, Pµ) is described by (1.8)–(1.9) and u is a
solution of the integral equation
(1.11) u(x) + Πx
Z τ
0
ψ[u(ξs)]ds = Πxf(ξτ ).
If ∂D is smooth and f is continuous, then (1.11) implies (1.5). Hence, (1.6) is a
solution of the problem (1.5).
Formulae (1.8) and (1.11) determine the probability distribution of XD for a
fixed D. Joint probability distributions of XD1 , · · · , XDn can be defined recursively
for every n by using the branching and Markov properties.
The following equations, similar to (1.8) and (1.11), describe the mass distri-
bution Xt at time t:
Pµ exph−f, Xti = exph−ut, µi,
(1.12)
ut(x) + Πx
Z t
0
ψ[ut−s(ξs)]ds = Πxf(ξt).
(1.13)
We cover both sets of equations, (1.8), (1.11) and (1.12), (1.13), by considering
exit measures (XQ, Pµ) for open subsets Q of the time-space S = R × Rd
and
measures µ on S. They satisfy the equations
Pµ exph−f, XQi = exph−u,µi,
(1.14)
u(r, x) + Πr,x
Z τ
r
ψ[u(s, ξs)]ds = Πr,xf(τ, ξτ )
(1.15)
where τ = inf{t : (t, ξt) /
∈ Q} is the first exit time from Q. Note that Xt = XS<t
where S<t = (−∞, t) × Rd
. If Q = (−∞, t) × D where D is a bounded smooth
domain 4
and if f is bounded and continuous, then u is a solution of a parabolic
equation
(1.16) u̇ +
1
2
∆u = ψ(u) in Q
such that u = f on ∂Q.
The maximal solution of the equation (1.3) can be described through the range
of X. This is the minimal closed set R which contains the support supp Xt for
all t. (It contains, a.s., supp XD for each D.) 5
For every open set D, a maximal
solution in D is given by
(1.17) u(x) = − log Px{R ⊂ D}.
4The name “smooth” is used for domains of class C2,λ (see section 6.1.3).
5Writing “a.s.” means Pµ-a.s. for all µ [or Πµ-a.s. for all µ in the case of a Brownian motion].](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-18-320.jpg)
![2. EXCEPTIONAL SETS IN ANALYSIS AND PROBABILITY 15
2. Exceptional sets in analysis and probability
2.1. Capacities. The most important class of exceptional sets in analysis are
sets of Lebesgue measure 0. The next important class are sets of capacity 0. A
capacity is a function C(B) ≥ 0 defined on all Borel sets. 6
It is not necessarily
additive but it is monotone increasing and continuous with respect to the monotone
increasing limits. For every B, C(B) is equal to the supremum of C(K) over all
compact sets K ⊂ B and it is equal to the infimum of C(O) over all open sets
O ⊃ B. [A more systematic presentation of Choquet’s capacities is given in section
10.3.2]
To every random closed set (F(ω), P) there corresponds a capacity
(2.1) C(B) = P{F ∩ B 6= ∅}.
Another remarkable class of capacities correspond to pairs (k, k·k) where k(x, y)
is a function on the product space E × Ẽ and k ·k is a norm in a space of functions
on E. The most important are the uniform norm
(2.2) kfk = sup
x
|f(x)|
and the Lα
(m)-norms
(2.3) kfkα =
Z
|f(x)|α
m(dx)
1/α
where 1 ≤ α ∞ and m is a measure on E. We assume that E and Ẽ are nice
metric spaces and that k(x, y) is positive valued, lower semicontinuous in x and
measurable in y. To every measure ν on Ẽ there corresponds a function
(2.4) Kν(x) =
Z
Ẽ
k(x, y) ν(dy)
on E. The capacity corresponding to (k, k · k) is defined on subsets B of Ẽ by the
formula
(2.5) C(B) = sup{ν(B) : ν is concentrated on B and kKνk ≤ 1}.
Our primary interest is not in capacities themselves but rather in the classes
of sets on which they vanish, and we say that two capacities are of the same type
if these classes coincide.
2.2. Exceptional sets for the Brownian motion. The Brownian motion
ξ in a domain D killed at the first exit time τ from D has a transition density
pt(x, y). If D is bounded, then
(2.6) g(x, y) =
Z ∞
0
pt(x, y)dt
is finite for x 6= y. We call g(x, y) Green’s function. The Green’s capacity corre-
sponds to the kernel g(x, y) and the uniform norm (2.2). 7
It is of the same type
as the capacity corresponding to
(2.7) g1(x, y) =
Z ∞
0
e−t
pt(x, y) dt
6And even on a larger class of analytic sets
7If d 1, then g(x, x) = ∞. Therefore Green’s capacity of a single point is equal to 0.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-19-320.jpg)
![16 1. INTRODUCTION
(and to the uniform norm). In the case d ≥ 3, it is also of the same type as the
capacity corresponding to the kernel |x − y|2−d
.
For a bounded set D, τ ∞ a.s. The range R of ξ is a continuous image of a
compact set [0, τ] and therefore, for every x ∈ D, (R, Πx) is a random closed set.
Consider the corresponding capacity
(2.8) Cx(B) = Πx{R ∩ B 6= ∅}.
A set B is called polar for ξ if Cx(B) = 0 for all x ∈ D B. This is equivalent to
the condition
(2.9) Πx{ξt ∈ B for some t} = 0 for all x ∈ D B
[in other words, a.s., ξ does not hit B ]. It is well-known (see, e.g., [Doo84]) that a
set B is polar if and only if its Green’s capacity is equal to 0. This gives an analytic
characterization of the class of polar sets.
2.3. Exceptional sets for the super-Brownian motion. We say that a
set B is polar for X if it is not hit by the range of X, that is if
(2.10) Px{R ∩ B 6= ∅} = 0 for all x /
∈ B.
In other words, B is polar, if, for all x /
∈ B, Capx
(B) = 0 where Capx
is the
capacity associated with a random closed set (R, Px). It was proved in [Dyn91c]
that all capacities Capx
are of the same type as the capacity determined by the
kernel (2.7) and the norm (2.3) (assuming that ψ is given by (1.4)).
It is clear from (1.17) that a closed set B is polar for X if and only if equation
(1.3) has only a trivial solution u = 0 in Rd
B. By the analytic result described
in section 1.2, a single point is polar if and only if d ≥ κα.
2.4. Exceptional boundary sets. Suppose that D is a bounded smooth
domain. Denote by γ(dy) the normalized surface area on ∂D. 8
If τ is the first exit
time of the Brownian motion ξ from D, then, for every Borel (or analytic) subset
Γ of ∂D,
(2.11) Πx{ξτ ∈ Γ} =
Z
Γ
k(x, y)γ(dy), x ∈ D
where k(x, y) is a strictly positive continuous function on D×∂D called the Poisson
kernel. Note that Πx{ξτ ∈ Γ} = 0 if and only if γ(Γ) = 0. In other words, the
capacity corresponding to a random closed set ({ξτ }, Πx) is of the same type as the
measure γ.
A class of exceptional boundary sets related to the super-Brownian motion X
is more interesting. It can be defined probabilistically in terms of the range RD of
X in D — the minimal closed subset supporting XD0 for all D0
⊂ D. Or it can be
introduced analytically via the capacity CPα corresponding to the Poisson kernel
k and the Lα
(m)-norm
(2.12) kfkα =
Z
D
|f(x)|α
m(dx)
1/α
.
Here m(dx) = dist(x, ∂D)dx. It is proved in Chapter 13 that
(2.13) Px{RD ∩ Γ 6= ∅} = 0 for all x ∈ D,
8This is a measure on ∂D determined by the Riemannian metric induced on ∂D by the
Euclidean metric in Rd. An explicit expression for γ is given in section 6.1.8.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-20-320.jpg)
![3. POSITIVE SOLUTIONS AND THEIR BOUNDARY TRACES 19
boundary set (described in section 2.4). If u corresponds to h and if tr h = ν, then
u can be considered as a solution of a generalized Dirichlet problem
∆u = ψ(u) in D,
u = ν on ∂D
(3.8)
(cf. (3.5)). It is natural to call ν the boundary trace of a moderate solution u.
3.5. Rough trace. For every u ∈ U(D) and for every closed subset B of ∂D
we define the sweeping QB(u) of u to B. In the case of a smooth domain, QB(u)
is the maximal solution dominated by u and equal to 0 on ∂D B. [The definition
is more complicated in the case of an arbitrary domain.]
The rough trace of u is a pair (Γ, ν) where Γ is a closed subset of ∂D and ν is
a Radon measure on O = ∂D Γ. Namely, Γ is the minimal closed set such that
QB(u) is moderate for all B disjoint from Γ. The measure ν is determined by the
condition: the restriction of ν to every B ⊂ O is equal to the trace of the moderate
solution QB(u).
The main results about the rough trace presented in Chapter 10 are:
A. Characterization of all pairs (Γ, ν) which are traces. [The principal condition
is that ν(B) = 0 for all exceptional boundary sets.]
B. Existence of the maximal solution with a given trace and an explicit prob-
abilistic formula for this solution.
Le Gall’s example (presented in section 3.5 of Chapter 10) shows that, in gen-
eral, infinitely many solutions can have the same rough trace.
3.6. Fine trace. Again this is a pair (Γ, ν) where Γ is a subset of ∂D and
ν is a measure on O = ∂D Γ. However Γ is not necessarily closed and ν is not
necessarily Radon measure.
Roughly speaking, the set Γ consists of points of the boundary near which u
rapidly tends to infinity. A precise definition can be formulated, both, in analytic
and probabilistic terms. Here we sketch a probabilistic approach based on the
concept of a Brownian motion in D conditioned to exit from D at a given point y
of the boundary. This stochastic process is described by a measure Πy
x on the space
of continuous paths which start at point x ∈ D and which are at y at the fist exit
time τ from D. Let f be a positive Borel function in D. We say that y is a point
of rapid growth of f if, for every x,
Z τ
0
f(ξs) ds = ∞ Πy
x-a.s.
We say that y is a singular point of a solution u if it is a point of rapid growth of
function ψ0
(u). We define Γ as the set of all singular points of u. To define the
measure ν, we consider all moderate solutions v ≤ u with the trace not charging Γ.
ν is the minimal measure such that, for every such v, tr v ≤ ν. We prove that:
A. A pair (Γ, ν) is a trace if and only if ν does not charge exceptional boundary
sets and if Γ contains all singular points of the following two solutions:
u∗
= sup {moderate v with the trace dominated by ν},
uΓ = sup {moderate v with the trace concentrated on Γ}](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-23-320.jpg)
![CHAPTER 2
Linear parabolic equations and diffusions
We introduce diffusions by using analytic results on fundamental solutions of
parabolic differential equations. A probabilistic approach to boundary value prob-
lems is based on the Perron method in PDEs. A central role is played by Poisson’s
and Green’s operators which we define in terms of diffusions. Fundamental concepts
of regular boundary points and of regular domains are also defined in probabilistic
terms.
1. Fundamental solution of a parabolic equation
1.1. Operator L. We work with functions u(r, x), r ∈ R, x ∈ E = Rd
on
(d + 1)-dimensional Euclidean space S = R × E. The first coordinate of a point
z ∈ S is interpreted as a time parameter. We write u̇ for ∂u
∂r and Diu for ∂u
∂xi
where
x1, . . ., xd are coordinates of x. Put Dij = DiDj.
Operator L is defined by the formula
(1.1) Lu(r, x) =
d
X
i,j=1
aij(r, x)Diju(r, x) +
d
X
i=1
bi(r, x)Diu(r, x)
where aij = aji. We assume that:
1.1.A. There exists a constant κ 0 such that
X
aij(r, x)titj ≥ κ
X
t2
i for all (r, x) ∈ S, t1, . . ., td ∈ R.
[ κ is called the ellipticity coefficient of L .]
1.1.B. aij and bi are bounded continuous and satisfy a Hölder’s type condition:
there exist constants 0 λ 1 and Λ 0 such that
|aij(r, x) − aij(s, y)| ≤ Λ(|x − y|λ
+ |r − s|λ/2
),
(1.2)
|bi(r, x) − bi(r, y)| ≤ Λ|x − y|λ
(1.3)
for all r, s ∈ R, x, y ∈ E.
For every interval I, we denote by SI or S(I) the slab I × E. We write St for
SI with I = (−∞, t) and we write Qt for the intersection of Q with St. . Writing
U b Q means that Q and U are open subsets of S, U is bounded and its closure
Ū is contained in Q. We use the name a sequence exhausting Q for a sequence of
open sets Qn ↑ Q such that Qn b Qn+1 for all n.
23](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-27-320.jpg)
![24 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS
1.2. Equation u̇ + Lu = 0. We investigate equation 1
(1.4) u̇ + Lu = 0 in Q.
Speaking about solutions of (1.4), we assume that the partial derivatives u̇, Diu, i =
1, . . ., d and Diju, i, j = 1, . . ., d are continuous. We denote C2
(Q) the class of
functions with this property.
Another class of functions plays a special role – continuous functions on Q that
are locally Hölder continuous in x uniformly in r. More precisely, we put u ∈ Cλ
(Q)
if u(r, x) is continuous on Q and if, for every compact Γ ⊂ Q, there exists a constant
ΛΓ such that
|u(r, x) − u(r, y)| ≤ ΛΓ|x − y|λ
for all (r, x), (r, y) ∈ Γ.
[λ (called Hölder’s exponent) satisfies the condition 0 λ 1.]
1.3. Fundamental solution. The following results are proved in the theory
of partial differential equations (see Chapter 1 in [Fri64] and section 4 in [IKO62]).
Theorem 1.1. There exists a unique continuous function p(r, x; t, y) on the set
{r t, x, y ∈ E} with the properties:
1.3.A. For every (t, y), the function u(r, x) = p(r, x; t, y) is a solution of
(1.5) u̇ + Lu = 0 in St.
1.3.B. For every t1 t2 and every δ 0, the function p(r, x; t, y) is bounded on
the set {t1 r t t2, t − r + |y − x| ≥ δ}.
1.3.C. If ϕ is continuous at a and bounded, then
Z
E
p(r, x; t, y)ϕ(y) dy → ϕ(a) as r ↑ t, x → a.
Function p is strictly positive and
Z
E
p(r, x; t, y) dy = 1 for all r t and all x;
(1.6)
Z
E
p(r, x; s, y)p(s, y; t, z) dy = p(r, x; t, z) for all r s t and all x, z.
(1.7)
Function p(r, x; t, y) is called a fundamental solution of equation (1.5).
We say that a function f is exp-bounded on B if supB |f(r, x)|e−β|x|2
∞ for
every β 0. (Clearly, all bounded functions are exp-bounded.)
We use the following properties of a fundamental solution.
1.3.1. If κ is the ellipticity coefficient of L, then, for every β κ,
(1.8)
p(r, x; t, y) ≤ C(t − r)−d/2
exp
−β|y − x|2
2(t − r)
for all t1 r t t2, x, y ∈ E
where the constant C depends on t1, t2 and β.
1This equation can be reduced by the time reversal r → −r to the equation u̇ = Lu which
is usually considered in the literature on partial differential equations. The form (1.4) is more
appropriate from a probabilistic point of view.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-28-320.jpg)
![2. DIFFUSIONS 25
1.3.2. If ϕ is an exp-bounded function on St = {t} × E, then
(1.9) u(r, x) =
Z
E
p(r, x; t, y)ϕ(y) dy
is exp-bounded on S[t0
, t) for every finite interval [t0
, t) and it satisfies equation
u̇ + Lu = 0 in St. If, in addition, ϕ is continuous, then, for every t0
t, u is a
unique exp-bounded solution of the problem
u̇ + Lu = 0 in S(t0
, t),
u = ϕ on St.
(1.10)
[Writing u = ϕ at z̃ ∈ ∂Q means u(z) → ϕ(z̃) as z ∈ Q tends to z̃.]
1.3.3. If ρ is a bounded Borel function on S(t0
, t) and if
(1.11) v(r, x) =
Z t
r
ds
Z
E
p(r, x; s, y)ρ(s, y) dy,
then Div are continuous on S(t0
, t) [and therefore v ∈ Cλ
[S(t0
, t)]]. If, in addition,
ρ ∈ Cλ
[S(t0
, t)], then v is a unique bounded solution of the problem
v̇ + Lv = −ρ in S(t0
, t),
v = 0 on St.
(1.12)
2. Diffusions
2.1. Continuous strong Markov processes. Here we describe a class of
Markov processes which contains all diffusions in a d-dimensional Euclidean space
E. 2
Imagine a particle moving at random in E. Suppose that the motion starts
at time r at a point x and denote ξt the state at time t ≥ r. The probability
that ξt belongs to a set B depends on r and x and we assume that it is equal to
R
B
p(r, x; t, y) dy. Moreover, we assume that, for every n = 1, 2, . . . and for all
r t1 · · · tn and all Borel sets B1, . . ., Bn,
(2.1) Probability of the event{ξt1 ∈ B1, . . ., ξtn ∈ Bn}
=
Z
B1
dy1 . . .
Z
Bn
dyn p(r, x; t1, y1)p(t1, y1; t2, y2) . . .p(tn−1, yn−1; tn, yn).
If the conditions (1.6)-(1.7) are satisfied, then the results of computation with
different n do no contradict each other and, by a Kolmogorov’s theorem 3
there
exists a probability measure Πr,x on the space of all paths in E starting at time
r which satisfies (2.1). We say that p(r, x; t, y) is the transition density of the
stochastic process (ξt, Πr,x). Sometimes the measures Πr,x can be defined on the
space of all continuous paths. For instance, this is possible for
(2.2) p(r, x; t, y) = [2π(t − r)]−d/2
exp
−
|x − y|2
2(t − r)
.
The corresponding continuous process is called the Brownian motion . Diffusions
also have continuous paths. (Their transition densities will be defined in section
2.2.)
2Basic facts on Markov processes are presented more systematically in the Appendix A.
3See [Kol33], Section III.4. Two proofs of Kolmogorov’s theorem are presented in [Bil95].](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-29-320.jpg)
![26 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS
We denote by Ωr the space of all continuous paths ω(t), t ∈ [r, ∞) in E. To
deal with a single space Ω, we introduce an extra point [ and we put ω(t) = [ for
ω ∈ Ωr and t r. We consider ξt as a function on Ω, namely, ξt(ω) = ω(t). The
birth time α is a function on Ω equal to r on Ωr. Measure Πr,x is concentrated
on the set {α = r, ξα = x}. For every interval I, the σ-algebra F(I) generated
by ξs, s ∈ I can be viewed as the class of all events determined by the behavior of
the path during I. Note that {α ≤ t} = {ξt ∈ E} belongs to F(I) for all I which
contain t. We use an abbreviation F≥t = F[t, ∞).
Every process (ξt, Πr,x) satisfies the following condition (which is called the
Markov property): events observable before and after time t are conditionally inde-
pendent given ξt. More precisely, if r t, A ∈ F[r, t] and B ∈ F≥t, then
(2.3) Πr,x(AB) =
Z
A
Πt,ξt (B)Πr,x(dω).
To simplify notation we write z for (r, x) and ηt for (t, ξt). Formula (2.3) implies
that for all X ∈ F[r, t] and every Y ∈ F≥t, 4
(2.4) Πz(XY ) = Πz(XΠηt Y ).
Diffusions satisfy a stronger condition called the strong Markov property. Roughly
speaking, it means that (2.4) can be extended to all stopping times τ. The defi-
nition of stopping times and their properties are discussed in the Appendix A. An
important class of stopping times are the first exit times. The first exit time from
an open set Q is defined by the formula
(2.5) τ(Q) = inf{t ≥ α : ηt /
∈ Q}.
[We put τ(Q) = ∞ if ηt ∈ Q for all t ≥ α.]We say that X is a pre-τ random
variable if X1τ≤t ∈ F≤t for all t.
In the Appendix A we give a general formulation of the strong Markov property,
we prove it for a wide class of Markov processes which includes all diffusions and
we deduce from it propositions 2.1.A–2.1.C — the only implications which we need
in this book.
2.1.A. Let ρ be a positive Borel function on S. For every stopping time τ and
every pre-τ X ≥ 0,
(2.6) ΠzX
Z ∞
τ
ρ(ηs) ds = ΠzXGρ(ητ )
where
(2.7) Gρ(z) = Πz
Z ∞
α
ρ(ηs) ds.
2.1.B. Let τ0
be the first exit time from an open set Q0
. Then for every stopping
time τ ≤ τ0
, for every pre-τ X ≥ 0 and for every Borel function f ≥ 0,
(2.8) ΠzX1Q0 (ητ )1τ0∞f(ητ0 ) = ΠzX1τ∞1Q0 (ητ )KQ0 f(ητ )
where
(2.9) KQ0 f(z) = Πz1τ0∞f(ητ0 ).
4Writing X ∈ F means that X ≥ 0 and X is measurable with respect to a σ-algebra F. It
can be proved (by using Theorem 1.1 in the Appendix A) that every X ∈ F≤t coincide Πr,x -a.e.
with a F[r, t]-measurable function. Therefore (2.4) holds for all X ∈ F≤t.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-30-320.jpg)
![2. DIFFUSIONS 27
[The value of η∞ is not defined. Instead of introducing in (2.8) and (2.9) factors
1τ0∞ and 1τ∞, we can agree to put f(η∞) = 0.]
2.1.C. Suppose that V is an open subset of S × S and τ is a stopping time. Put
(2.10) σt = inf{u ≥ α : u t, (ηt, ηu) /
∈ V }.
If στ ∞ Πz-a.s. for all z, then, for every pre-τ function X ≥ 0 and every Borel
function f ≥ 0,
(2.11) ΠzXf(ηστ ) = ΠzXF(ητ )
where
(2.12) F(t, y) = Πt,yf(ησt ).
2.2. L-diffusion. An L-diffusion is a continuous strong Markov process with
transition density p(r, x; t, y) which is a fundamental solution of (1.5). The existence
of such a process is proved in Chapter 5 of [Dyn65].
Note that
(2.13) Πr,xϕ(ξt) =
Z
E
p(r, x; t, y)ϕ(y) dy.
It follows from Fubini’s theorem that
(2.14) Πr,x
Z t
r
ρ(s, ξs) ds =
Z t
r
ds
Z
E
p(r, x; s, y)ρ(s, y) dy.
Therefore, under the conditions on ϕ and ρ formulated in 1.3.2–1.3.3,
(2.15) u(r, x) = Πr,xϕ(ξt)
is a solution of the problem (1.10) and
(2.16) v(r, x) = Πr,x
Z t
r
ρ(s, ξs) ds
is a solution of the problem (1.12).
2.3. Martingales associated with L-diffusions. Martingales are one of
new tools contributed to analysis by probability theory. 5
The following theorem
establishes a link between martingales and parabolic differential equations.
Theorem 2.1. Suppose f ∈ C2
(S(t1, t2)) is exp-bounded on S(t1, t2) and that
ρ = ˙
f + Lf is bounded and belongs to Cλ
(S(t1, t2)). Then, for every r ∈ (t1, t2)
and every x ∈ E,
(2.17) Yt = f(ηt) −
Z t
r
ρ(ηs) ds, t ∈ (r, t2)
is a martingale with respect to F[r, t] and Πr,x.
Proof. First, we prove that, for all r t and all x,
(2.18) w(r, x) = Πr,x[f(ηt) − f(ηr) −
Z t
r
ρ(ηs) ds]
is equal to 0. Indeed, w = u−f −v where u is defined by (2.15) with ϕ(x) = f(t, x)
and v is defined by (2.16). It follows from 1.3.2–1.3.3, that w is an exp-bounded
5See the Appendix A for basic facts on martingales.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-31-320.jpg)
![28 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS
solution of problem (1.10) with ϕ = 0. Such a solution is unique and therefore
w = 0.
For every t, Yt is measurable relative to F[r, t] and Πr,x|Yt| ∞. We need to
prove that, for all r ≤ t0
t and for every bounded F[r, t0
]-measurable X,
(2.19) Πr,xX(Yt − Yt0 ) = 0.
Note that
Yt − Yt0 = f(ηt) − f(ηt0 ) −
Z t
t0
ρ(ηs) ds
is F≥t0-measurable. By the Markov property (2.4),
Πr,xX(Yt − Yt0 ) = Πr,xXΠt0,ξt0 (Yt − Yt0 )
and (2.19) holds because, by (2.18), Πt0,y(Yt − Yt0 ) = 0 for all y.
Corollary. Suppose that U b Q and let τ be the first exit time from U. If
f ∈ C2
(Q) and if ρ = ˙
f + Lf ∈ Cλ
(Q), then
(2.20) Πr,xf(ητ ) = f(r, x) + Πr,x
Z τ
r
( ˙
f + Lf)(ηs) ds.
Proof. Since U is bounded, it is contained in SI for some finite interval I.
There exists a bounded function of class C2
(SI) which coincides with f on Ū and
therefore we can assume that f is defined on SI and that it satisfies the conditions
of Theorem 2.1. The martingale Yt given by (2.17) is continuous and τ is bounded
Πr,x-a.s. By Theorem 4.1 in the Appendix A, Πr,xYτ = Πr,xYr which implies
(2.20).
3. Poisson operators and parabolic functions
3.1. Poisson operators. The Poisson operator corresponding to an open set
Q is defined by the formula
(3.1) KQf(z) = Πz1τ∞f(ητ )
where τ is the first exit time from Q (cf. formula (2.9)). Note that KQf = f on
Qc
. It follows from 2.2.1.B that, for every U b Q and every f ≥ 0,
(3.2) KU KQf = KQf.
3.2. Parabolic functions. We say that a continuous function u in Q is par-
abolic if, for every open set U b Q,
(3.3) KU u = u in U.
The following lemma is an immediate implication of Corollary to Theorem 2.1.
Lemma 3.1. Every solution u of the equation
(3.4) u̇ + Lu = 0 in Q
is a parabolic function in Q.
We say that a Borel subset T of ∂Q is total if, for all z ∈ Q,
Πz{τ ∞, ητ ∈ T } = 1.
In particular, ∂Q is total if and only if Πz{τ = ∞} = 0 for all z ∈ Q. [This
condition holds, for instance, if Q ⊂ St with a finite t.] If ∂Q is not total, then
there exist no total subsets of ∂Q.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-32-320.jpg)
![3. POISSON OPERATORS AND PARABOLIC FUNCTIONS 29
Lemma 3.2. Suppose T is a total subset of ∂Q. If u is bounded and continuous
on Q ∪ T and if it is parabolic in Q, then
(3.5) KQu = u in Q.
Proof. Consider a sequence Qn exhausting Q. The sequence τn = τ(Qn) is
monotone increasing. Denote its limit by σ. For almost all ω, σ ≤ τ ∞ and
ησ ∈ T . Therefore σ = τ(Q). We get (3.5) by passing to the limit in the equation
u(z) = Πzu(ητn ).
Lemma 3.3. Suppose that parabolic functions un converge to u at every point
of Q. If un are locally uniformly bounded, then u is also parabolic.
Proof. If U b Q, then un are uniformly bounded on Ū. By passing to the
limit in the equation KU un = un, we get KU u = u.
3.3. Poisson operator corresponding to a cell. Subsets of S of the form
C = (a0, b0) × (a1, b1) × · · · × (ad, bd) are called (open) cells. Points of ∂C with
the first coordinate equal to a0 form the bottom B of C. Clearly, T = ∂C B is a
total subset. We denote it ∂rC. A basic result proved in every book on parabolic
equations 6
implies that, if f is a bounded continuous function on ∂rC, then there
exists a continuous function u on C ∪ ∂rC such that
u̇ + Lu = 0 in C,
u =f on ∂rC.
(3.6)
It follows from Lemmas 3.1 and 3.2 that u = KCf. Note that KC is continuous
with respect to the bounded convergence. It follows from the multiplicative systems
theorem (Theorem 1.1 in the Appendix A) that these two properties characterize
KC. This provides a purely analytic definition of KC.
A particular class of cells is defined by the formula
C(z, β) = {z0
: d̃(z, z0
) β}
where
d̃(z, z0
) = maxi |xi − x0
i| for z = (x0, . . ., xd), z0
= (x0
0, . . ., x0
d).
3.4. Superparabolic and subparabolic functions. A lower semicontinu-
ous function u is called superparabolic if, for every open set U b Q,
(3.7) KU u ≤ u in U.
A function u is called subparabolic if −u is superparabolic.
Lemma 3.4. Suppose that u is a bounded below lower semicontinuous function
in Q and that (3.7) holds for every cell C b Q. Then u is superparabolic and,
moreover, (3.7) holds for all U ⊂ Q.
Proof. For U = S, the relation (3.7) is satisfied because its left side is 0. If
U 6= S, then
d(z) = inf
z0∈∂U
d̃(z, z0
) ∞
for all z. Put
V = {(z, z̃) : ˜
d(z, z̃)
1
2
d(z)}
6See, e.g., Chapter 3, section 4 in [Fri64] or Chapter V, section 2 in [Lie96].](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-33-320.jpg)
![30 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS
and consider the function σt defined by the formula (2.10). The stopping times
τ0 = α, τn+1 = στn for n ≥ 0
are finite and τ0 ≤ τ1 ≤ · · · ≤ τn ≤ . . .. It follows from 2.1.C that Πzu(ητn+1 ) =
ΠzF(ητn ) where F(z) = Πzu(ητ1 ). If (3.7) is satisfied for cells, then F(z) ≤ u(z).
Hence, Πzu(ητn+1 ) ≤ Πzu(ητn ) and, by induction, Πzu(ητn ) ≤ Πzu(ητ0 ) = u(z).
We have d̄(ητn+1 , ητn) = d(ητn )/2. If τ is the limit of τn, then, on the set {τ ∞},
ητn → ητ and therefore 0 = d̃(ητ , ητ ) = d(ητ )/2. We conclude that {τ ∞} ⊂
{ητ ∈ ∂U} ⊂ {τ = τ(U)}. By the definition of the lower semicontinuity, on the set
{τ ∞}, u(ητ ) ≤ lim inf u(ητn ). Therefore, by Fatou’s lemma,
Πz1τ∞u(ητ ) ≤ Πz1τ∞ lim inf u(ητn ) ≤ lim inf Πz1τ∞u(ητn ) ≤ u(z).
Lemma 3.5. Suppose that w is superparabolic in Q and bounded below. Let T
be a total subset of ∂Q. If, for every z̃ ∈ T ,
(3.8) lim inf w(z) ≥ 0 as z → z̃,
then w ≥ 0 in Q.
Proof. Let τ = τ(Q). It follows from Lemma 3.4 that Πz1τ∞w(ητ ) ≤ w(z).
Condition (3.8) implies that w(ητ ) ≥ 0 Πz-a.s. on {τ ∞}. Hence w(z) ≥ 0.
3.5. The Perron solution. The following analytic results are proved, for
instance, in [Lie96], Chapter III, section 4. 7
Let f be a bounded Borel function
on ∂Q. A bounded below superparabolic function w is in the upper Perron class
U+ for f, if
(3.9) lim inf
z→z̃
w(z) ≥ f(z̃) for all z̃ ∈ ∂Q.
Analogously, a bounded above subparabolic function v is in the lower Perron
class U− for f, if
lim sup
z→z̃
w(z) ≤ f(z̃) for all z̃ ∈ ∂Q.
It follows from Lemma 3.5 that v ≤ w for every v ∈ U− and every w ∈ U+.
Since f is bounded, all sufficiently big constants are in U+ and all sufficiently small
constants are in U−. Therefore functions of class U− are uniformly bounded from
above and functions of class U+ are uniformly bounded from below.
It is proved that the infimum u of all functions w ∈ U+ coincides with the
supremum of all functions v ∈ U−. Moreover, u is a solution of the equation (3.4).
It is called the Perron solution corresponding to f.
Theorem 3.1. If Q is a bounded open set and f is a bounded Borel function
on ∂Q, then u = KQf is the Perron solution corresponding to f.
Proof. Let w ∈ U+ and let τ be the first exit time from Q. Consider a
sequence Qn exhausting Q and the corresponding first exit times τn. We have
w(z) ≥ Πzw(ητn ) and, by condition (3.9) and Fatou’s lemma,
w(z) ≥ lim inf Πzw(ητn ) ≥ Πz lim inf w(ητn ) ≥ Πzf(ητ ).
Similarly, if v ∈ U−, then v(z) ≤ Πzf(ητ ).
7See also [Doo84], 1.XVIII.1. There only the case L = ∆ is considered but the arguments
can be modified to cover a general L.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-34-320.jpg)
![3. POISSON OPERATORS AND PARABOLIC FUNCTIONS 31
Corollary. A function u is parabolic in Q if and only if it is a solution of
(3.4).
3.6. Smooth superparabolic functions. The improved Maximum prin-
ciple. We wish to prove:
3.6.A. If u ∈ C2
(Q) and if
(3.10) u̇ + Lu ≤ 0 in Q,
then U is superparabolic in Q.
This follows immediately from (2.20) if u̇ + Lu ∈ Cλ
(Q). To eliminate this
restriction, we use:
3.6.B. Suppose that C is a cell and u ∈ C2
(C) satisfies the conditions
u̇ + Lu ≥ 0 in C,
lim supu(z) ≤ 0 as z → z̃ for all z̃ ∈ ∂rC.
Then u ≤ 0 in C.
[This proposition is proved in any book on parabolic PDEs (for instance, in
Chapter 2 of [Fri64] or in Chapter II of [Lie96]).]
To prove 3.6.A we consider an arbitrary cell C b Q. As we know, v = KCu is a
solution of the problem (3.6) with f equal to the restriction of u to ∂rC. Therefore
w = v − u satisfies conditions ẇ + Lw ≥ 0 in C and w(z) → 0 as z → z̃ ∈ ∂rC. By
3.6.B, w ≤ 0 in C. Hence, KCu ≤ u. By Lemma 3.4, u is superparabolic.
3.6.C. [The improved maximum principle.] Let T be a total subset of ∂Q. If
v ∈ C2
(Q) is bounded above and satisfies the condition
(3.11) v̇ + Lv ≥ 0 in Q
and if, for every z̃ ∈ T ,
(3.12) lim supv(z) ≤ 0 as z → z̃,
then v ≤ 0 in Q.
Indeed, by 3.6.A, u = −v is superparabolic and, by Lemma 3.5, u ≥ 0.
3.7. Superparabolic functions and supermartingales.
Proposition 3.1. Suppose u is a positive lower semicontinuous superparabolic
function in Q and τ = τ(Q). Then, for every r, x,
Xt = 1tτ u(ηt)
is a supermartingale on [r, ∞) relative to F[r, t] and Πr,x.
Proof. Note that σ = τ ∧ t is the first exit time from Q ∩ St. Since σ ∞,
by Lemma 3.4, for every s t,
Πs,xu(ησ) ≤ u(s, x).
Since {σ τ} = {σ = t}, we have
Πs,xXt = Πs,x1σ=tu(ηt) = Πs,x1σ=tu(ησ) ≤ u(s, x).](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-35-320.jpg)
![32 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS
Let τr be the first after r exit time from Q. If r s, then {τr s} ∈ F[r, s].
Clearly, τr = τ Πr,x-a.s. If A ∈ F[r, s], then {A, s τr} ∈ F[r, s] and, by the
Markov property (2.4),
Z
A,sτr
Xt dΠr,x =
Z
A,sτr
Πηs Xt dΠr,x ≤
Z
A,sτr
u(ηs) dΠr,x.
For s t, Xt1sτr = Xt Πr,x-a.s. and therefore
R
A Xt dΠr,x ≤
R
A Xs dΠr,x. Since
Xt is F[r, t]-measurable and Πr,x-integrable, it is a supermartingale.
4. Regular part of the boundary
4.1. Regular points. A point z̃ = (r̃, x̃) of ∂Q is called regular if, for every
t r̃,
(4.1) Πz̃{ηs ∈ Q for all s ∈ (r̃, t)} = 0.
.
Theorem 4.1. Let τ be the first exit time from Q. If a point z̃ = (r̃, x̃) ∈ ∂Q
is regular then, for every t r̃,
(4.2) Πz{τ t} → 0 as z ∈ Q tends to z̃.
Proof. 1◦
. Fix t and put, for every r ≤ s t, A(s, t) = {ηu ∈ Q for u ∈
(s, t)} and qr
s(x) = Πr,xA(s, t). Note that qr
r(x) = Πr,x{τ t} for (r, x) ∈ Q.
Therefore the conditions (4.1) and (4.2) are equivalent to the conditions qr̃
r̃(x̃) = 0
and qr
r(x) → 0 as (r, x) → (r̃, x̃).
2◦
. By the Markov property of ξ, for all r ≤ s t,
qr
s(x) = Πr,xΠs,ξs A(s, t) =
Z
E
p(r, x; s, y)qs
s(y) dy.
3◦
. It follows from 2◦
and 1.3.2 that qr
s(x) is continuous in (r, x) for r s.
Therefore, for every ε 0, there exists a neighborhood U of (r̃, x̃) such that
|qr
s(x) − qr̃
s(x̃)| ε for all (r, x) ∈ U.
4◦
. Clearly, qr
s(x) ↓ qr
r (x) as s ↓ r.
5◦
. Suppose r̃ t. If (r̃, x̃) is regular, then, by 1◦
, qr̃
r̃(x̃) = 0 and, by 4◦
, for
every ε 0, there exists s ∈ (r̃, t) such that qr̃
s(x̃) ε. By 3◦
, if (r, x) ∈ U, then
qr
r(x) ≤ qr
s(x) ≤ qr̃
s(x̃) + |qr
s(x) − qr̃
s(x̃)| 2ε.
By 1◦
, this implies (4.2).
Remark. The converse to Theorem 4.1 is also true: (4.2) implies (4.1). We
do not use this fact. In an elliptic setting, it is proved in Chapter 13 of [Dyn65].
The role of condition (4.2) is highlighted by the following theorem:
Theorem 4.2. If (4.2) holds at z̃ ∈ ∂Q and if a bounded function f on ∂Q is
continuous at z̃, then
(4.3) KQf(z) → f(z̃) as z → z̃.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-36-320.jpg)
![4. REGULAR PART OF THE BOUNDARY 33
Informally, we have the following implications:
(4.4) {z = (r, x) ∈ Q is close to z̃ = (r̃, x̃)} =⇒ {τ is close to r}
=⇒ {ητ is close to z and therefore close to z̃}
=⇒ {f(ητ ) is close to f(z̃)} =⇒ {Πzf(ητ ) is close to f(z̃)}.
A rigorous proof is based on the following lemma.
Lemma 4.1. Fix t ∈ R and put
(4.5) Dr = sup
rst
|ηs − ηr|.
For every ε 0, there exits δ 0 such that
(4.6) Πr,x{Dr ε} ε
for all x and all r ∈ (t − δ, t).
Proof. If ξs = (ξ1
s , . . ., ξd
s ), then
Dr ≤ t − r +
d
X
1
Di
r
where
Di
r = sup
rst
|ξi
s − ξi
r|.
To prove the lemma it is sufficient to show that, for every ε 0, there exists δ 0
such that
(4.7) Πr,x{Di
r ε} ε
for all x and all r ∈ (t − δ, t). Each function fi(r, x) = xi is exp-bounded on S and
ρi = ˙
fi + Lfi = bi (a coefficient in (1.1)). The conditions of Theorem 2.1 hold for
fi on every interval (t1, t2). Therefore, for every r ∈ (t1, t2),
Y i
s = ξi
s −
Z s
r
ρi(ηu) du, s ∈ (r, t2)
is a martingale relative to F[r, s], Πr,x. Choose (t1, t2) which contains [r, t]. By
Kolmogorov’s inequality (see section 4.4 in the Appendix A)
(4.8) Πr,x{ sup
rst
|Y i
s − Y i
r | δ} ≤ δ−2
Πr,x|Y i
t − Y i
r |2
.
If |ρi| ≤ c, then
(4.9) Di
r ≤ sup
rst
|Y i
s − Y i
r | + c(t − r).
On the other hand, since (A + B)2
≤ 2A2
+ 2B2
for all A, B, we have
(4.10) Πr,x|Y i
t − Y i
r |2
≤ 2Πr,x|ξi
t − ξi
r|2
+ 2c2
(t − r)2
.
The bound (1.8) implies that
(4.11)
Πr,x(ξi
t − ξi
r)2
=
Z
E
p(r, x; t, y)(yi − xi)2
dy ≤
Z
E
p(r, x; t, y)|y − x|2
dy ≤ C(t − r)
where C is a constant [depending on t1, t2]. The bound (4.7) follows from (4.8)–
(4.11).](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-37-320.jpg)
![34 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS
Proof of Theorem 4.2. Let z = (r, x), z̃ = (r̃, x̃). For every t r̃,
Πr,x{|ητ − z| ε} ≤ Πr,x{τ ≥ t} + Πr,x{Dr ε}.
By Lemma 4.1, for every ε 0, there exists δ 0 such that
Πr,x{Dr ≥ ε} ε
for all x and all r ∈ (t − δ, t). Choose t ∈ (r̃, r̃ + δ/2). Then every r ∈ (r̃ − δ/2, r̃)
belongs to (t − δ, t).
By Theorem 4.1,
Πr,x{τ ≥ t} ε
in a neighborhood U of z̃. Suppose that r ∈ (t − δ, t) and z ∈ U. Then
Πr,x{|ητ − z| ε} ≤ 2ε.
Let V be the intersection of U with the ε-neighborhood of z̃. If z ∈ V , then
Πr,x{|ητ − z̃| 2ε} ≤ Πr,x{|ητ − z| ε} ≤ 2ε.
Suppose that N is an upper bound for |f| and let |f(z)−f(z̃)| ε for z ∈ V . Then
Πr,x|f(ητ ) − f(z̃)| ≤ 2NΠr,x{|ητ − z̃| δ} + ε ≤ (2N + 1)ε in V
which implies (4.3).
Theorem 4.3. Suppose Q is bounded and all points of a total subset T of ∂Q
are regular. If a function f is bounded and continuous on T , then u = KQf is a
unique bounded solution of the problem
u̇ + Lu = 0 in Q,
u = f on T .
(4.12)
Proof. Put f = 0 on ∂QT . By Theorems 3.1 and 4.2, u = KQf is a solution
of the problem (4.12). Clearly, u is bounded. For an arbitrary bounded solution v
of (4.12), by Lemmas 3.1 and 3.2, v = KQv = KQf = u.
We say that a function u is a barrier at z̃ if there exists a neighborhood U of
z̃ such that u ∈ C2
(U) ∩ C(Ū) and
(4.13) u̇ + Lu ≤ 0 in Q ∩ U, u(z̃) = 0, u 0 on Ū ∩ Q̄ except z̃.
Lemma 4.2. The condition (4.2) holds if there exists a barrier u at z̃. 8
Proof. Put V = Q ∩ U. For every t r̃, the infimum β of u on the set
∂V ∩ S≥t is strictly positive. Let τ = τ(V ). By Chebyshev’s inequality and (3.1),
(4.14) Πz{τ t} ≤ Πz{u(ητ ) ≥ β} ≤ Πzu(ητ )/β = KV u(z)/β.
By 3.6.A, u is superparabolic in V . Denote by f the restriction of u to ∂V . Clearly,
u belongs to the upper Perron class for f. By Theorem 3.1, KV f is the correspond-
ing Perron solution. Hence, KV u = KV f ≤ u, and (4.14) implies (4.2).
By constructing a suitable barrier, we prove that (4.2) holds if z̃ can be touched
from outside by a ball. More precisely, we have the following test.
8The existence of a barrier is also a necessary condition for the regularity of z̃. (See, e.g.,
[Lie96], Lemma 3.23 or [Dyn65], Theorem 13.6.)](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-38-320.jpg)
![4. REGULAR PART OF THE BOUNDARY 35
Theorem 4.4. The property (4.2) holds at z̃ = (r̃, x̃) ∈ ∂Q if there exists
z0
= (r0
, x0
) with x0
6= x̃ such that |z − z0
| |z̃ − z0
| for all z ∈ Q̄ sufficiently close
to z̃ and different from z̃. In other words, z̃ is the only common point of three sets:
Q̄, a closed ball centered at z0
and a neighborhood of z̃.
Proof. We claim that, if ε = |z̃ − z0
| and if p is sufficiently large, then
u(z) = ε−2p
− |z − z0
|−2p
,
is a barrier at z̃. Clearly, u(z̃) = 0. There exists a neighborhood U of z̃ such that,
for all z ∈ Ū ∩ Q̄, |z − z0
| ε and therefore u(z) 0. We have
u̇ + Lu = A[−(p + 1)B + C]
where
A = 2p|z − z0
|−2(p+2)
, B =
X
aij(xi − x0
i)(xj − x0
j),
C = |z − z0
|2
X
[aii + bi(xi − x0
i)].
Note that B ≥ κ|x − x0
|2
where κ is the ellipticity coefficient of L. Since aij and bi
are bounded, we see that, for sufficiently large p, u̇ + Lu ≤ 0 in a neighborhood of
z̃ assuming that x̃ 6= x0
.
4.2. Regular open sets. We denote by ∂regQ the set of all regular points
of ∂Q and by ∂rQ the set of all interior (relative to ∂Q) points of ∂regQ. We say
that Q is regular if ∂regQ contains a total subset of ∂Q. A smaller class of strongly
regular open sets is defined by the condition: ∂rQ is total in ∂Q.
For a cell C, ∂rC coincides with the set introduced in section 3.3. This set is
relatively open in ∂C and therefore cells are strongly regular open sets.
Note that the following conditions are equivalent: (a) B is a relatively open
subset of A; (b) B = A ∩ O where O is an open subset of S; (c) A = B ∪ F where
F is a closed subset of S. Therefore, if Bi is a relatively open subset of Ai, i = 1, 2,
then B1 ∩B2 is relatively open in A1 ∩A2 and B1 ∪B2 is relatively open in A1 ∪A2.
Lemma 4.3. If U is strongly regular, then Q = U ∩ Q1 is strongly regular for
every open set Q1 such that Ū ∩ ∂Q1 ⊂ ∂rQ1.
Proof. The boundary ∂Q is the union of three sets A1 = ∂U ∩ Q1, A2 = U ∩
∂Q1 and A3 = ∂U ∩∂Q1. Sets B1 = ∂rU ∩Q1, B2 = U ∩∂rQ1 and B3 = ∂rU ∩∂rQ1
are relatively open in, respectively, A1, A2, A3 and therefore T = B1 ∪ B2 ∪ B3 is
relatively open in ∂Q. Every point of T is regular in ∂Q. It remains to show that T
is total in ∂Q. Let τ be the first exit time from Q and let z ∈ Q. Since ∂rU is total
in ∂U and ∂rU ∩ (A1 ∪ A3) ⊂ B1 ∪ B3, we have {ητ ∈ A1 ∪ A3} ⊂ {ητ ∈ B1 ∪ B3}
Πz-a.s. On the other hand, A2 ⊂ ∂rQ1 and therefore A2 = B2 and {ητ ∈ A2} =
{ητ ∈ B2}.
Now we introduce an important class of simple open sets. We start from closed
cells [a0, b0] × [a1, b1] × · · · × [ad, bd]. We call finite unions of closed cells simple
compact sets. We define a simple open set as the collection of all interior points of a
simple compact set . The boundary ∂C of a cell C = [a0, b0]×[a1, b1]×· · ·×[ad, bd]
consists of 2(d + 1) d-dimensional faces. We distinguish two horizontal faces: the
top {b0} × [a0, b0] × [a1, b1] × · · ·× [ad, bd] and the bottom {a0} × [a0, b0] × [a1, b1] ×
· · · × [ad, bd]. We call the rest side faces.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-39-320.jpg)
![36 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS
Theorem 4.5. Every simple open set is strongly regular. For an arbitrary open
set Q, there exists a sequence of simple open sets exhausting Q.
In the proof we use the following observations.
4.2.A. Let H be a (d − 1)-dimensional affine subspace of Rd
[that is the set of
x = (x1, . . ., xd) such that a1x1 + · · · + adxd = c for some constants a1, . . ., ad
not all equal to 0]. Then for all r t, x ∈ Rd
, Πr,x{ξt ∈ H} = 0. If H is a
(d − 2)-dimensional affine subspace, then Πr,x{ξt ∈ H for some t r} = 0.
The first part holds because the probability distribution of ξt is absolutely
continuous with respect to the Lebesgue measure. We leave the second part as an
exercise for a reader.
4.2.B. If F is a (d − 1)-dimensional face of a cell C, then
Πr,x{(t, ξt) ∈ F for some t r} = 0 for all (r, x) ∈ S.
This follows easily from 4.2.A.
Proof of Theorem 4.5. Every compact simple set A can be represented as the
union of closed cells C1, . . ., Cn such that the intersection of every two distinct cells
Ci, Cj is either empty or it is a common face of both cells. Let Q be the set of all
interior points of A. Note that ∂Q = ∪N
1 Fk where F1, . . ., FN are d-dimensional
cells which enter the boundary of exactly one of Ci. Clearly, the set F0
k of all points
of Fk that do not belong to any (d − 1)-dimensional face of any Ci is open in ∂Q.
By 4.2.B, to prove that Q is strongly regular, it is sufficient to show that, for every
k, either Πz{ητ ∈ F0
k } = 0 for all z ∈ Q or F0
k ⊂ ∂regQ. Clearly, the first case takes
place if Fk is the bottom of Ci. If Fk is the top of Ci, then, obviously, F0
k ⊂ ∂regQ.
If Fk is a side face, then F0
k ⊂ ∂regQ by Theorem 4.4.
It remains to construct sets Qn. It is easy to reduce the general case to the case
of a bounded Q. Suppose that Q is bounded. Put εn = (d + 1)1/2
2−n
. Consider a
partition of S = Rd+1
into cells with vertices in the lattice 2−n
Zd+1
and take the
union An of all cells whose εn-neighborhood are contained in Q. The set Qn of all
interior points of An
is a simple open set. Clearly, the sequence Qn exhaust Q.
For every two sets A, B, we denote by d(A, B) the infimum of d(a, b) = |a − b|
over all a ∈ A, b ∈ B.
Suppose that Q is an open set and Γ is a closed subset of ∂Q. We say that a
sequence of open sets Qn ↑ Q is a (Q, Γ)-sequence if Qn are bounded and strongly
regular and if
(4.15) Q̄n ↑ Q̄ Γ; d(Qn, Q Qn+1) 0.
Lemma 4.4. A (Q, Γ)-sequence exists if Γ contains all irregular points of ∂Q.
Proof. By Theorem 4.5, there exists a sequence of strongly regular open sets
Un exhausting S Γ. If Γ contains all irregular points of ∂Q, then, by Lemma 4.3,
sets Qn = Un ∩ Q are strongly regular.
Note that Q̄n ⊂ Q̄ and Q̄n ∩ Γ ⊂ Ūn ∩ Γ ⊂ Un+1 ∩ Γ = ∅. Hence Q̄n ⊂ Q̄ Γ.
If K is a compact set disjoint from Γ, then K ⊂ Un for some n. Let x ∈ Q̄ Γ.
For sufficiently small δ 0, K = {y : |y −x| ≤ δ} is disjoint from Γ. If xm → x and
xm ∈ Q, then, for sufficiently large m, xm ∈ Un ∩ Q = Qn. Hence x ∈ Q̄n. This
proves the first part of (4.15). The second part holds because Qn ⊂ Ūn, QQn+1 ⊂
Uc
n+1 and d(Ūn, Uc
n+1) 0.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-40-320.jpg)
![5. GREEN’S OPERATORS AND EQUATION u̇ + Lu = −ρ 37
5. Green’s operators and equation u̇ + Lu = −ρ
5.1. Parts of a diffusion. A part ˜
ξ of a diffusion ξ in an arbitrary open set
Q ⊂ S is obtained by killing ξ at the first exit time τ from Q. More precisely, we
consider
˜
ξt = ξt for t ∈ [α, τ),
= † for t ≥ τ
where † – “the cemetery” – is an extra point (not in E). The state space at time
t is the t-section Qt = {x : (t, x) ∈ Q} of Q. We will show that ˜
ξ = (˜
ξt, Πr,x) is a
Markov process with the transition density
(5.1) pQ(r, x; t, y) = p(r, x; t, y) − Πr,xp(τ, ξτ ; t, y) for x ∈ Qr, y ∈ Qt.
[We set p(r, x; t, y) = 0 for r ≥ t.]
Theorem 5.1. For every Borel function f ≥ 0 on Q,
(5.2) Πr,x1tτ f(t, ξt) =
Z
Qt
pQ(r, x; t, y)f(t, y) dy for all x ∈ Qr.
Moreover, for every n = 1, 2, . . . and for all r t1 · · · tn, x ∈ Qr and Borel
sets B1, . . ., Bn,
(5.3) Πr,x{τ tn, ξt1 ∈ B1, . . ., ξtn ∈ Bn}
=
Z
B1
dy1 . . .
Z
Bn
dynpQ(r, x; t1, y1)pQ(t1, y1; t2, y2) . . .pQ(tn−1, yn−1; tn, yn).
We have:
pQ(r, x; t, y) ≥ 0 for all r t, x ∈ Qr, y ∈ Qt;
(5.4)
Z
Qt
pQ(r, x; t, y) dy ≤ 1 for all r t and all x ∈ Qr;
(5.5)
(5.6)
Z
Qs
pQ(r, x; s, y)pQ(s, y; t, z) dy = pQ(r, x; t, z)
for all r s t and all x ∈ Qr, z ∈ Qt.
Proof. If we set f = 0 outside Q, then
(5.7) Πr,x1τ=tf(t, ξt) = 0
because f(τ, ξτ ) = 0. Therefore
(5.8) Πr,x1tτ f(t, ξt) = u(r, x) − v(r, x)
where
u(r, x) = Πr,xf(t, ξt),
v(r, x) = Πr,x1τtf(t, ξt).
By (1.9) and (2.15),
(5.9) u(s, x) =
Z
E
p(s, x; t, y)f(t, y)dy.
By 2.1.B (applied to τ0
= t),
(5.10) v(r, x) = Πr,x1τtf(t, ξt) = Πr,x1τtF(τ, ξτ )](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-41-320.jpg)
![5. GREEN’S OPERATORS AND EQUATION u̇ + Lu = −ρ 39
There exists a simple relation between pQ and pQ0 for Q0
⊂ Q:
(5.12) pQ0 (r, x; t, y) = pQ(r, x; t, y) − Πr,xpQ(τ0
, ξτ0 ; t, y)
where τ0
= τ(Q0
). Indeed, put f(r, x) = p(r, x; t, y) and denote by fQ, fQ0 the
functions obtained in a similar way from pQ, pQ0 . By (5.1), fQ = f − KQf and, by
(3.2), KQ0 fQ = KQ0 f − KQf. Hence, fQ − fQ0 = −KQf + KQ0 f = KQ0 fQ.
5.3. Green’s operators. Green’s operator in an arbitrary open set Q is de-
fined by the formula
(5.13) GQρ(r, x) = Πr,x
Z τ
r
ρ(s, ξs) ds
(cf. (2.7)). By (5.2),
(5.14) GQρ(r, x) =
Z ∞
r
ds
Z
Qs
pQ(r, x; s, y)ρ(s, y)dy.
If Q0
⊂ Q, then, by (5.14) and (5.12),
(5.15) GQ = GQ0 + KQ0 GQ.
5.3.A. Suppose that Q ⊂ SI = I × E, I is a finite interval and ρ is bounded.
Then function w = GQρ belongs to Cλ
(Q). If ρ ∈ Cλ
(Q), then w ∈ C2
(Q) and it
is a solution of the equation
(5.16) ẇ + Lw = −ρ in Q.
If ρ is bounded and if z̃ is a regular point of ∂Q, then
(5.17) w(z) → 0 as z → z̃.
Proof. Note that w = v − KQv where v is given by (1.11). Since KQv is
parabolic in Q, the first part of 5.3.A follows from 1.3.3.
If z = (r, x) and if N is an upper bound of |ρ|, then, for every ε 0, |w(z)| ≤
N[(t − r)Πz{τ r + ε} + ε] and therefore (5.17) follows from Theorem 4.1.
5.3.B. Let τ be the first exit time from an arbitrary open set Q. If ρ ≥ 0 and
w = GQρ is finite at a point z ∈ Q, then
(5.18) lim
t↑τ
w(ηt) = 0 Πz-a.s.
Proof. Let z = (r, x). We can assume that ρ ≥ 0. We prove that Mt =
1tτ w(ηt), t ∈ [r, ∞) is a supermartingale relative to F[r, t], Πr,x. To this end, we
consider a bounded positive F[r, t]-measurable function X and we note that, by the
Markov property (2.4),
Πr,xX1tτ
Z τ
t
ρ(s, ξs) ds = Πr,xX1tτ Πt,ξt
Z τ
t
ρ(s, ξs) ds = Πr,xX1tτ w(t, ξt).
Hence Πr,xXMt ≤ Πr,xXMs for r ≤ s ≤ t. Since Mt is F[r, t]-measurable and
Πr,x-integrable, our claim is proved.
Since Mt is right continuous, a limit Mτ− as t ↑ τ exists Πr,x-a.s. (see 4.3.C in
the Appendix A). Suppose Qn exhaust Q and let τn be the first exit time from Qn.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-43-320.jpg)
![40 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS
By (5.15), w = GQn ρ + KQn w. Since GQn ρ ↑ GQρ, we conclude that KQn w ↓ 0
and
Πr,xMτ− = Πr,x limw(τn, ξτn ) ≤ limΠr,xw(τn, ξτn ) = limKQn w = 0.
5.3.C. If Q, ρ and f are bounded and if f is continuous on ∂regQ, then
(5.19) v = GQρ + KQf
is a solution of the problem
v̇ + Lv = −ρ in Q,
v = f on ∂regQ.
(5.20)
This follows from 5.3.A and Theorems 3.1 and 4.2.
5.3.D. Let ρ be bounded. A function w is a solution of equation (5.16) if and
only if w is locally bounded and, for every U b Q,
(5.21) w = GU ρ + KU w.
Proof. Suppose that w satisfies (5.16). By Theorem 4.5, for an arbitrary
U there exists a sequence of regular open sets Un ↑ U. Since w is bounded and
continuous on Ū, we have KUn w → KU w. Also GUn ρ → GUρ. Therefore it is
sufficient to prove (5.21) for a regular U.
By 5.3.A and Theorem 4.3, u = GUρ + KU w − w is a solution of the problem
u̇ + Lu = 0 in U,
u = 0 on ∂regU.
(5.22)
By 3.6.C, u = 0.
If w satisfies (5.21) and is bounded on Ū, then the equation (5.16) holds on U
by 5.3.A and Theorem 3.1.
5.3.E. Suppose that solutions wn of (5.16) converge to w at every point of Q.
If wn are locally uniformly bounded, then w also satisfies (5.16).
This follows from 5.3.D (cf. the proof of Lemma 3.3).
6. Notes
Our treatment of diffusions is in spirit of the book [Dyn65]. However, in this
book only time-homogeneous case was considered. Inhomogeneous diffusions were
covered in [Dyn93]. In particular, one can found there a probabilistic formula for
the Perron solutions, the improved maximum principle and an approximation of
arbitrary domains by simple domains. A concept of strongly regular domains was
introduced in [Dyn98a]. This class of domains plays a special role in the theory
of semilinear partial differential equations (see, Chapter 5).
A fundamental monograph of Doob [Doo84] contains the most complete pre-
sentation of the connections between the Brownian motion and classical potential
theory related to the Laplace equation. Bibliographical notes in [Doo84] should be
consulted for the early history of this subject. A special role in the book is played
by martingale theory. Much of this theory was created by Doob.
Martingale are the principal tool used by Stroock and Varadhan to develop a
new approach to diffusions. A construction of diffusions by solving a martingale
problem is presented in their monograph [SV79].](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-44-320.jpg)
![6. NOTES 41
A direct construction of the paths of diffusions by solving stochastic differential
equations is due to Itô [Itô51]. A modern presentation of Itô’s calculus and its
applications is given in the books of Ikeda and Watanabe [IW81] and Rogers and
Williams [RW87].](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-45-320.jpg)
![CHAPTER 3
Branching exit Markov systems
In this chapter we introduce a general model — BEM systems — which is
the basis for the theory of superprocesses and, in particular, superdiffusions to
be developed in the next chapters. A BEM system describes a mass distribution
of a random cloud started from a distribution µ and frozen at the exit from Q.
Mathematically, this is a family X of random measures (XQ, Pµ) in a space S. The
parameter Q takes values in a class of subsets of S, µ is a measure on S, XQ is a
function of ω ∈ Ω and Pµ is a probability measure on Ω. A Markov property is
defined with the role of “past” and “future” played by Q0
⊂ Q and Q00
⊃ Q. [This
definition can be applied to a parameter Q taking values in any partially ordered
set.] We consider systems which combine the Markov property with a branching
property which means, heuristically, an absence of interaction between any parts
of the random cloud described by X.
We start from historical roots of the concept of branching. Then we introduce
branching particle systems (they were described on a heuristic level in Chapter 1)
and we use them to motivate a general definition of BEM systems. The transition
operators VQ play a role similar to the role of the transition functions in the theory
of Markov processes. We investigate properties of these operators and we show how
a BEM system can be constructed starting from a family of operators VQ. At the
end of the chapter some basic properties of BEM systems are proved.
1. Introduction
1.1. Simple models of branching. The first probabilistic model of branch-
ing appeared in 1874 in the problem of the family name extinction posed by Francis
Galton and solved by H. W. Watson [WG74]. Galton’s motivation was to evaluate
a conjecture that the extinction of prominent families is more likely than the ex-
tinction of ordinary ones. He suggested to start from probabilities pn for a man to
have n sons evaluated by the demographical data for the general population. The
problem consisted in computation of the probability of extinction after k genera-
tions. Watson’s solution contained an error but he introduced a tool of fundamental
importance for the theory of branching. The principal observation was: if
ϕ(z) =
∞
X
0
pnzk
is the generating function for the number of sons, then the generating function
ϕk for the number of descendants in the k-th generation can be evaluated by the
recursive formula
(1.1) ϕk+1 = ϕ(ϕk).
43](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-47-320.jpg)
![44 3. BRANCHING EXIT MARKOV SYSTEMS
The Galton-Watson model and its modifications found many applications in biology,
physics, chemistry... 1
A model of branching with a continuous time parameter was suggested in 1947
in [KD47] (an output of a Kolmogorov’s seminar held at Moscow University in
1946-47). Consider a particle system and assume that a single particle produces,
during time interval (r, t), k = 0, 1, 2, . . . particles with probability pk(r, t). Gener-
ating functions
ϕ(r, t; z) =
∞
X
0
pk(r, t)zk
satisfy the condition
(1.2) ϕ(r, t; z) = ϕ(r, s; ϕ(s, t; z)) for r s t.
Suppose that
pk(r − h, r) = ak(r)h + o(h) for k 6= 1,
p1(r − h, r) = 1 + a1(r)h + o(h)
as h ↓ 0 and let
(1.3) Φ(r; z) =
∞
X
0
ak(r)zk
.
We arrive at a differential equation
(1.4)
∂ϕ(r, t; z)
∂r
+ Φ(r, ϕ(r, t; z)) = 0 for r t
with a boundary condition
(1.5) Φ(r, t; z) → z as r ↑ t.
Equation (1.4) and a linear equation u̇ + Lu = 0 considered in Chapter 2 are
particular cases of a semilinear parabolic equation
(1.6) u̇ + Lu = ψ(u).
A probabilistic approach to (1.6) is based on a model which involves both L-diffusion
and branching.
1.2. Exit systems associated with branching particle systems. Con-
sider a system of particles moving in E according to the following rules:
(1) The motion of each particle is described by a right continuous strong Markov
process ξ.
(2) A particle dies during time interval (t, t + h) with probability kh + o(h),
independently on its age.
(3) If a particle dies at time t at point x, then it produces n new particles with
probability pn(t, x).
(4) The only interaction between the particles is that the birth time and place
of offspring coincide with the death time and place of their parent.
[Assumption (2) implies that the life time of every particle has an exponential
probability distribution with the mean value 1/k.]
We denote by Pr,x the probability law corresponding to a process started at
time r by a single particle located at point x. Suppose that particles stop to move
1More on an early history of the branching processes can be found in [Har63].](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-48-320.jpg)
![46 3. BRANCHING EXIT MARKOV SYSTEMS
Condition 1.3.A (we call it the continuous branching property) implies that
PµZ =
Y
Pµn Z
for all Z ∈ Z if µn, n = 1, 2, . . . and µ =
P
µn belong to M.
A family X is called an exit system if:
1.3.B. For all µ ∈ M and Q ∈ O,
Pµ{XQ(Q) = 0} = 1.
1.3.C. If µ ∈ M and µ(Q) = 0, then
Pµ{XQ = µ} = 1.
Finally, we say that X is a branching exit Markov [BEM] system, if XQ ∈ M
for all Q ∈ O and if, in addition to 1.3.A–1.3.C, we have:
1.3.D. [Markov property.] Suppose that Y ≥ 0 is measurable with respect to the
σ-algebra F⊂Q generated by XQ0 , Q0
⊂ Q and Z ≥ 0 is measurable with respect to
the σ-algebra F⊃Q generated by XQ00 , Q00
⊃ Q.
Then
(1.10) Pµ(Y Z) = Pµ(Y PXQ Z).
It follows from the principles (1)-(4) stated at the beginning of section 1.2 that
conditions 1.3.A–1.3.D hold for the systems of random measures associated with
branching particle systems. For them S = R × E, M = N(S) and O is a class of
open subsets of S.
1.4. Transition operators. Let X = (XQ, Pµ), Q ∈ O, µ ∈ M be a family
of random measures. Denote by B the set of all bounded positive BS-measurable
functions. Operators VQ, Q ∈ O acting on B are called the transition operators of
X if, for every µ ∈ M and every Q ∈ O,
(1.11) Pµe−hf,XQ i
= e−hVQ(f),µi
.
If X is a branching system, then (1.11) follows from the formula
(1.12) VQ(f)(y) = − logPye−hf,XQ i
for f ∈ B.
In this chapter we establish sufficient conditions for operators VQ to be transi-
tion operators of a branching exit Markov system. In the next chapter we study a
special class of BEM systems which we call superprocesses.
A link between operators VQ and a BEM system X is provided by a family of
transition operators of higher order VQ1,...,Qn . We call it a V-family.
2. Transition operators and V-families
2.1. Transition operators of higher order. Suppose that
(2.1) Pµ exp[−hf1, XQ1 i − · · · − hfn, XQn i],
= exp[−hVQ1,...,Qn (f1, . . ., fn), µi]](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-50-320.jpg)
![2. TRANSITION OPERATORS AND V-FAMILIES 47
for all µ ∈ M, f1, . . ., fn ∈ B and Q1, . . ., Qn ∈ O. Then we say that operators
VQ1,...,Qn are the transition operators of order n for X. Condition (2.1) is equivalent
to the assumption that X is a branching system and that
(2.2) VQ1,...,Qn (f1, . . ., fn)(y) = − log Py exp[−hf1, XQ1 i − · · · − hfn, XQn i],
f1, . . ., fn ∈ B, y ∈ S.
[For n = 1, formulae (2.1)–(2.2) coincide with (1.11)–(1.12).]
We use the following abbreviations. For every finite subset I = {Q1, . . ., Qn}
of O, we put
XI = {XQ1 , . . ., XQn },fI = {f1, . . ., fn},
hfI , XIi =
n
X
i=1
hfi, XQi i.
(2.3)
In this notation, formulae (2.2) and (2.1) can be written as
(2.4) VI(fI )(y) = − logPye−hfI ,XIi
and
(2.5) Pµe−hfI ,XI i
= e−hVI (fI ),µi
.
If X satisfies condition 1.3.C, then:
2.1.A. For every Qi ∈ I, VI(fI ) = fi +VIi (fIi ) on Qc
i where Ii is the set obtained
from I by dropping Qi.
Indeed,
hfI , XIi = hfi, XQi i + hfIi , XIi i
and hfi, XQi i = fi(y) Py-a.s. if y ∈ Qc
i .
For a branching exit system X, the Markov property 1.3.D is equivalent to:
2.1.B. If Q ⊂ Qi for all Qi ∈ I, then
(2.6) VQVI = VI.
Formula (2.6) can be rewritten in the form
(2.7) VI(fI) = VQ[1Qc VI (fI)] for all fI.
Proof. It follows from (2.5) that
e−hVQVI(fI ),µi
= Pµe−hVI (fI ),XQi
= PµPXQ e−hfI ,XI i
.
If Q ⊂ Qi for all Qi ∈ I, then hfI , XIi ∈ F⊃Q and 1.3.D implies that the right side
is equal to
Pµe−hfI ,XI i
= e−hVI (fI ),µi
.
Hence (2.6) follows from 1.3.D. By 1.3.B, for every F, the value of VQ(F) does not
depend on the values of F on Q. Therefore (2.7) and (2.6) are equivalent.
To deduce 1.3.D from 2.1.B, it is sufficient to prove (1.10) for
Y = e−hfI ,XI i
, Z = e−hfĨ ,XĨ i
where I = {Q1, . . ., Qn}, ˜
I = {Q̃1, . . ., Q̃m} with Qi ⊂ Q ⊂ Q̃j. Note that Y Z ∈ Z.
By 1.3.A, the same is true for Y PXQ Z. Therefore (1.10) will follow from 1.3.A if](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-51-320.jpg)
![48 3. BRANCHING EXIT MARKOV SYSTEMS
we check that it holds for all µ = δy. We use the induction in n. The condition
(2.6) implies
(2.8) PµZ = PµPXQ Z.
Hence, (1.10) holds for n = 0. Suppose it holds for n−1. If y ∈ Qc
i , then, by 1.3.C,
Py{Y = e−fi(y)
Yi} = 1 where Yi = e−hfIi
,XIi
i
, and we have
PyY Z = e−fi(y)
PyYiZ = e−fi(y)
Py(YiPXQ Z) = Py(Y PXQ Z)
by the induction hypothesis. Hence (1.10) holds for δy with y not in the intersection
QI of Qi ∈ I. For an arbitrary y, by (2.8), PyY Z = PyPXQI
Y Z. By 1.3.B, XQI is
concentrated, Py-a.s. on Qc
I and therefore
PXQI
Y Z = PXQI
(Y PXQ Z).
We conclude that
PyY Z = PyPXQI
(Y PXQ Z) = Py(Y PXQ Z).
Transition operators of order n for a BEM system can be expressed through
transition operators of order n − 1 by the formulae
(2.9) VI(fI ) = fi + VIi (fIi ) on Qc
i for every Qi ∈ I,
(2.10) VI (fI) = VQI [1Qc
I
VI(fI )] where QI is the intersection of all Qi ∈ I.
Formula (2.9) (equivalent to 2.1.A) defines the values of VI(fI ) on Qc
I. Formula
(2.10) follows from (2.7). It provides an expression for all values of VI(fI ) through
its values on Qc
I.
Conditions (2.9)–(2.10) can be rewritten in the form
(2.11) VI = VQI ṼI
where
(2.12) ṼI(fI ) =
(
fi + VIi (fIi ) on Qc
i ,
0 on QI.
2.2. Properties of VQ. We need the following simple lemma.
Lemma 2.1. Let Y be a positive random variable and let 0 ≤ c ≤ ∞. If
Pe−λY
≤ e−λc
for all λ 0, then P{Y ≥ c} = 1. If, in addition Pe−Y
= e−c
,
then P{Y = c} = 1.
Proof. If c = ∞, then, P-a.s., e−λY
= 0 and therefore Y = ∞. If c ∞,
then Pe−λ(Y −c)
≤ 1 and, by Fatou’s lemma, P{ lim
λ→∞
e−λ(Y −c)
} ≤ 1. Hence, P{Y ≥
c} = 1. The second part of the lemma follows from the first one.
Theorem 2.1. Transition operators of an arbitrary system of finite random
measures X satisfy the condition:
2.2.A. For all Q ∈ O,
(2.13) VQ(fn) → 0 as fn ↓ 0.
A branching system X is a branching exit system if and only if:](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-52-320.jpg)
![3. FROM A V-FAMILY TO A BEM SYSTEM 49
2.2.B.
VQ(f) = VQ( ˜
f) if f = ˜
f on Qc
.
2.2.C. For every Q ∈ O and every f ∈ B,
VQ(f) = f on Qc
.
Proof. 1◦
. Property 2.2.A is obvious. It is clear that 1.3.B implies 2.2.B and
1.3.C implies 2.2.C.
2◦
. If 2.2.B holds, then VQ(1Q) = VQ(0) = 0 and therefore Pye−XQ(Q)
= 1
which implies 1.3.B.
3◦
. It follows from 2.2.C and (1.11) that, if µ(Q) = 0, then, for all f ∈ B and
all λ 0,
Pµe−λhf,XQ i
= e−λhf,µi
and, by Lemma 2.1,
(2.14) hf, XQi = hf, µi Pµ-a.s.
Since there exists a countable family of f ∈ B which separate measures, 1.3.C
follows from (2.14).
2.3. V-families. We call a collection of operators VI a V-family if it satisfies
conditions (2.9)–(2.10) [equivalent to (2.11)–(2.12)] and 2.2.A. We say that a V-
family and a system of random measures correspond to each other if they are
connected by formula (2.1).
Theorem 2.2. Suppose that operators VQ, Q ∈ O satisfy conditions 2.2.A–
2.2.C and the condition
2.3.A. For all Q ⊂ Q̃ ∈ O,
VQVQ̃ = VQ̃.
Then there exists a V-family {VI} such that VI = VQ for I = {Q}.
Proof. Denote by |I| the cardinality of I. For |I| = 1, operators VI are
defined. Suppose that VI, subject to conditions (2.9)–(2.10), are already defined
for |I| n. For |I| = n, we define VI by (2.9)–(2.10). This is not contradictory
because
fi + VIi (fIi ) = fj + VIj (fIj ) = fi + fj + VIij (fIij ) on Qc
i ∩ Qc
j.
By 2.2.B it is legitimate to define VI (fI) on QI by (2.10).
3. From a V-family to a BEM system
3.1. P-matrices and N-matrices. First, we prepare some algebraic and an-
alytic tools.
Suppose that a symmetric n×n matrix (aij) satisfies the condition: for all real
numbers t1, . . ., tn,
n
X
i,j=1
aijtitj ≥ 0.
In algebra, such matrices are called positive semidefinite. Some authors (e.g.,
[BJR84]) use the name positive definite. We resolve this controversy by using
a short name a P-matrix.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-53-320.jpg)
![50 3. BRANCHING EXIT MARKOV SYSTEMS
We need another class of matrices which are called negative definite in [BJR84].
[This is inconsistent with the common usage in algebra where negative definite
means (−1)× positive definite.] We prefer again a short name. We call an n × n
symmetric matrix an N-matrix if
n
X
i,j=1
aijtitj ≤ 0
for every n ≥ 2 and all t1, . . ., tn ∈ R such that
P
ti = 0.
The following property of these classes is obvious:
3.1.A. The classes P and N are closed under entry wise convergence. Moreover,
they are convex cones in the following sense: if (B, B, η) is a measure space, if aij(b)
is a P-matrix (N-matrix) for all b ∈ B and if aij(b) are η-integrable, then
aij =
Z
aij(b)η(db)
is also a P-matrix (respectively, an N-matrix).
Here are some algebraic properties of both classes.
(i) A matrix (aij) is a P-matrix if and only if it has a representation
aij =
m
X
k=1
qikqjk
where m ≤ n.
This follows from the fact that a quadratic form is positive semidefinite if and
only if it can be transformed by a linear transformation to the sum of m ≤ n
squares.
(ii) If (aij) and (bij) are P-matrices, then so is the matrix cij = aijbij.
Indeed, by using (i), we get
X
ij
cijtitj =
X
k
X
ij
bij(qikti)(qjktj) ≥ 0.
(iii) If (aij) is a P-matrix, then cij = eaij
is also a P-matrix.
This follows from (ii) and 3.1.A.
(iv) Suppose that a (n + 1) × (n + 1) matrix (aij)n
0 and an n × n-matrix (bij)n
1
are connected by the formula
(3.1) bij = −aij + ai0 + a0j − a00, i, j = 1, . . ., n.
Then, for all t0, . . ., tn such that t0 + · · · + tn = 0,
n
X
i,j=0
aijtitj = −
n
X
i,j=1
bijtitj.
Therefore (aij) is an N-matrix if and only if (bij) is a P-matrix.
Now we can prove the following proposition:
3.1.B. A matrix (aij) belongs to class N if and only if cij(λ) = e−λaij
is a
P-matrix for all λ 0.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-54-320.jpg)
![[173] See the Preface to my book, called, The Crucifying of the
World.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-70-320.jpg)
![those that concur in the injury, being accessories, are bound to
satisfy. As, 1. Those that teach or command another to do it. 2.
Those who send a commission, or authorize another to do it. 3.
Those who counsel, exhort, or persuade another to do it. 4. Those
who by consenting are the causes of it. 5. Those who co-operate
and assist in the injury knowingly and voluntarily. 6. Those who
hinder it not when they could and were obliged to do it. 7. Those
who make the act their own, by owning it, or consenting afterward.
8. Those who will not reveal it afterward, that the injured party may
recover his own, when they are obliged to reveal it. But a secret
consent which no way furthered the injury, obligeth none to
restitution, but only to repentance; because it did no wrong to
another, but it was a sin against God.
Quest. IV. To whom must restitution or satisfaction be made?
Answ. 1. To the true owner, if he be living and to be found, and it
can be done. 2. If that cannot be, then to his heirs, who are the
possessors of that which was his. 3. If that cannot be, then to God
himself, that is, to the poor, or unto pious uses; for the possessor is
no true owner of it; and therefore where no other owner is found,
he must discharge himself so of it, to the use of the highest and
principal Owner, as may be most agreeable to his will and interest.
[174]
Quest. V. What restitution should he make who hath dishonoured
his governors or parents?
Answ. He is bound to do all that he can to repair their honour, by
suitable means; and to confess his fault, and crave their pardon.
Quest. VI. How must satisfaction be made for slanders, lies, and
defaming of others?
Answ. By confessing the sin, and unsaying what was said, not only
as openly as it was spoken, but as far as it is since carried on by
others, and as far as the reparation of your neighbour's good name
requireth, if you are able.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-78-320.jpg)
![Direct. III. When really you are bound to restitution or
satisfaction, stick not at the cost or suffering, be it never so great,
but be sure to deal faithfully with God and conscience. Else you will
keep a thorn in your hearts, which will smart and fester till it be out:
and the ease of your consciences will bear the charge of your
costliest restitution.
Direct. IV. If you be not able in your lifetime to make restitution,
leave it in your wills as a debt upon your estates; but never take it
for your own.
Direct. V. If you are otherwise unable to satisfy, offer your labour
as a servant to him to whom you are indebted; if at least by your
service you can make him a compensation.
Direct. VI. If you are that way unable also, beg of your friends to
help you, that charity may enable you to pay the debt.
Direct. VII. But if you have no means at all of satisfying, confess
the injury and crave forgiveness, and cast yourself on the mercy of
him whom you have injured.
[174] Heb. v. 23; 1 Sam xii. 3; Neh. v. 11; Numb. v. 8; Luke xix.
8.](https://image.slidesharecdn.com/442309-250518142223-c35af43d/85/Diffusions-Superdiffusions-And-Pdes-Dynkin-Eb-83-320.jpg)
Diffusions Superdiffusions And Pdes Dynkin Eb
- 1. Diffusions Superdiffusions And Pdes Dynkin Eb download https://ebookbell.com/product/diffusions-superdiffusions-and- pdes-dynkin-eb-884618 Explore and download more ebooks at ebookbell.com
- 2. Here are some recommended products that we believe you will be interested in. You can click the link to download. Diffusions Superdiffusions And Partial Differential Equations E B Dynkin https://ebookbell.com/product/diffusions-superdiffusions-and-partial- differential-equations-e-b-dynkin-1296458 Diffusions And Waves Henryk Gzyl Auth https://ebookbell.com/product/diffusions-and-waves-henryk-gzyl- auth-4204052 Affine Diffusions And Related Processes Simulation Theory And Applications Aurlien Alfonsi https://ebookbell.com/product/affine-diffusions-and-related-processes- simulation-theory-and-applications-aurlien-alfonsi-5087504 Degenerate Diffusions Initial Value Problems And Local Regularity Theory Ems Tracts In Mathematics Panagiota Daskalopoulos And Carlos E Kenig https://ebookbell.com/product/degenerate-diffusions-initial-value- problems-and-local-regularity-theory-ems-tracts-in-mathematics- panagiota-daskalopoulos-and-carlos-e-kenig-1707428
- 3. Random Walks And Diffusions On Graphs And Databases An Introduction 1st Edition Philippe Blanchard https://ebookbell.com/product/random-walks-and-diffusions-on-graphs- and-databases-an-introduction-1st-edition-philippe-blanchard-2265718 Hybrid Switching Diffusions Properties And Applications 1st Edition G George Yin https://ebookbell.com/product/hybrid-switching-diffusions-properties- and-applications-1st-edition-g-george-yin-4194432 Point Processes And Jump Diffusions An Introduction With Finance Applications Tomas Bjrk https://ebookbell.com/product/point-processes-and-jump-diffusions-an- introduction-with-finance-applications-tomas-bjrk-42014350 Functionals Of Multidimensional Diffusions With Applications To Finance 1st Edition Jan Baldeaux https://ebookbell.com/product/functionals-of-multidimensional- diffusions-with-applications-to-finance-1st-edition-jan- baldeaux-4319714 Nonlocal And Nonlinear Diffusions And Interactions New Methods And Directions Cetraro Italy 2016 1st Edition Jos Antonio Carrillo Et Al Eds https://ebookbell.com/product/nonlocal-and-nonlinear-diffusions-and- interactions-new-methods-and-directions-cetraro-italy-2016-1st- edition-jos-antonio-carrillo-et-al-eds-6789190
- 5. Diffusions, superdiffusions and partial differential equations E. B. Dynkin Department of Mathematics, Cornell University, Malott Hall, Ithaca, New York, 14853
- 7. Contents Preface 7 Chapter 1. Introduction 11 1. Brownian and super-Brownian motions and differential equations 11 2. Exceptional sets in analysis and probability 15 3. Positive solutions and their boundary traces 17 Part 1. Parabolic equations and branching exit Markov systems 21 Chapter 2. Linear parabolic equations and diffusions 23 1. Fundamental solution of a parabolic equation 23 2. Diffusions 25 3. Poisson operators and parabolic functions 28 4. Regular part of the boundary 32 5. Green’s operators and equation u̇ + Lu = −ρ 37 6. Notes 40 Chapter 3. Branching exit Markov systems 43 1. Introduction 43 2. Transition operators and V-families 46 3. From a V-family to a BEM system 49 4. Some properties of BEM systems 56 5. Notes 58 Chapter 4. Superprocesses 59 1. Definition and the first results 59 2. Superprocesses as limits of branching particle systems 63 3. Direct construction of superprocesses 64 4. Supplement to the definition of a superprocess 68 5. Graph of X 70 6. Notes 74 Chapter 5. Semilinear parabolic equations and superdiffusions 77 1. Introduction 77 2. Connections between differential and integral equations 77 3. Absolute barriers 80 4. Operators VQ 85 5. Boundary value problems 88 6. Notes 91 3
- 8. 4 CONTENTS Part 2. Elliptic equations and diffusions 93 Chapter 6. Linear elliptic equations and diffusions 95 1. Basic facts on second order elliptic equations 95 2. Time homogeneous diffusions 100 3. Probabilistic solution of equation Lu = au 104 4. Notes 106 Chapter 7. Positive harmonic functions 107 1. Martin boundary 107 2. The existence of an exit point ξζ− on the Martin boundary 109 3. h-transform 112 4. Integral representation of positive harmonic functions 113 5. Extreme elements and the tail σ-algebra 116 6. Notes 117 Chapter 8. Moderate solutions of Lu = ψ(u) 119 1. Introduction 119 2. From parabolic to elliptic setting 119 3. Moderate solutions 123 4. Sweeping of solutions 126 5. Lattice structure of U 128 6. Notes 131 Chapter 9. Stochastic boundary values of solutions 133 1. Stochastic boundary values and potentials 133 2. Classes Z1 and Z0 136 3. A relation between superdiffusions and conditional diffusions 138 4. Notes 140 Chapter 10. Rough trace 141 1. Definition and preliminary discussion 141 2. Characterization of traces 145 3. Solutions wB with Borel B 147 4. Notes 151 Chapter 11. Fine trace 153 1. Singularity set SG(u) 153 2. Convexity properties of VD 155 3. Functions Ju 156 4. Properties of SG(u) 160 5. Fine topology in E0 161 6. Auxiliary propositions 162 7. Fine trace 163 8. On solutions wO 165 9. Notes 166 Chapter 12. Martin capacity and classes N1 and N0 167 1. Martin capacity 167 2. Auxiliary propositions 168 3. Proof of the main theorem 170
- 9. CONTENTS 5 4. Notes 172 Chapter 13. Null sets and polar sets 173 1. Null sets 173 2. Action of diffeomorphisms on null sets 175 3. Supercritical and subcritical values of α 177 4. Null sets and polar sets 179 5. Dual definitions of capacities 182 6. Truncating sequences 184 7. Proof of the principal results 192 8. Notes 198 Chapter 14. Survey of related results 203 1. Branching measure-valued processes 203 2. Additive functionals 205 3. Path properties of the Dawson-Watanabe superprocess 207 4. A more general operator L 208 5. Equation Lu = −ψ(u) 209 6. Equilibrium measures for superdiffusions 210 7. Moments of higher order 211 8. Martingale approach to superdiffusions 213 9. Excessive functions for superdiffusions and the corresponding h-transforms 214 10. Infinite divisibility and the Poisson representation 215 11. Historical superprocesses and snakes 217 Appendix A. Basic facts on Markov processes and martingales 219 1. Multiplicative systems theorem 219 2. Stopping times 220 3. Markov processes 220 4. Martingales 224 Appendix B. Facts on elliptic differential equations 227 1. Introduction 227 2. The Brandt and Schauder estimates 227 3. Upper bound for the Poisson kernel 228 Epilogue 231 1. σ-moderate solutions 231 2. Exceptional boundary sets 231 3. Exit boundary for a superdiffusion 232 Bibliography 235 Subject Index 243 Notation Index 245
- 11. Preface Interactions between the theory of partial differential equations of elliptic and parabolic types and the theory of stochastic processes are beneficial for, both, prob- ability theory and analysis. At the beginning, mostly analytic results were used by probabilists. More recently, the analysts (and physicists) took inspiration from the probabilistic approach. Of course, the development of analysis, in general, and of theory of partial differential equations, in particular, was motivated to a great ex- tent by the problems in physics. A difference between physics and probability is that the latter provides not only an intuition but also rigorous mathematical tools for proving theorems. The subject of this book is connections between linear and semilinear differ- ential equations and the corresponding Markov processes called diffusions and su- perdiffusions. A diffusion is a model of a random motion of a single particle. It is characterized by a second order elliptic differential operator L. A special case is the Brownian motion corresponding to the Laplacian ∆. A superdiffusion describes a random evolution of a cloud of particles. It is closely related to equations involving an operator Lu − ψ(u). Here ψ belongs to a class of functions which contains, in particular ψ(u) = uα with α > 1. Fundamental contributions to the analytic theory of equations (0.1) Lu = ψ(u) and (0.2) u̇ + Lu = ψ(u) were made by Keller, Osserman, Brezis and Strauss, Loewner and Nirenberg, Brezis and Véron, Baras and Pierre, Marcus and Véron. A relation between the equation (0.1) and superdiffusions was established, first, by S. Watanabe. Dawson and Perkins obtained deep results on the path behavior of the super-Brownian motion. For applying a superdiffusion to partial differential equations it is insufficient to consider the mass distribution of a random cloud at fixed times t. A model of a superdiffusion as a system of exit measures from time- space open sets was developed in [Dyn91c], [Dyn92], [Dyn93]. In particular, a branching property and a Markov property of such system were established and used to investigate boundary value problems for semilinear equations. In the present book we deduce the entire theory of superdiffusion from these properties. We use a combination of probabilistic and analytic tools to investigate positive solutions of equations (0.1) and (0.2). In particular, we study removable singulari- ties of such solutions and a characterization of a solution by its trace on the bound- ary. These problems were investigated recently by a number of authors. Marcus and Véron used purely analytic methods. Le Gall, Dynkin and Kuznetsov combined 7
- 12. 8 PREFACE probabilistic and analytic approach. Le Gall invented a new powerful probabilistic tool — a path-valued Markov process called the Brownian snake. In his pioneer- ing work he used this tool to describe all solutions of the equation ∆u = u2 in a bounded smooth planar domain. Most of the book is devoted to a systematic presentation (in a more general setting, with simplified proofs) of the results obtained since 1988 in a series of papers of Dynkin and Dynkin and Kuznetsov. Many results obtained originally by using superdiffusions are extended in the book to more general equations by applying a combination of diffusions with purely analytic methods. Almost all chapters involve a mixture of probability and analysis. Exceptions are Chapters 7 and 9 where the probability prevails and Chapter 13 where it is absent. Independently of the rest of the book, Chapter 7 can serve as an introduction to the Martin boundary theory for diffusions based on Hunt’s ideas. A contribution to the theory of Markov processes is also a new form of the strong Markov property in a time inhomogeneous setting. The theory of parabolic partial differential equations has a lot of similarities with the theory of elliptic equations. Many results on elliptic equations can be easily deduced from the results on parabolic equations. On the other hand, the analytic technique needed in the parabolic setting is more complicated and the most results are easier to describe in the elliptic case. We consider a parabolic setting in Part 1 of the book. This is necessary for constructing our principal probabilistic model — branching exit Markov systems. Superprocesses (including superdiffusions) are treated as a special case of such sys- tems. We discuss connections between linear parabolic differential equations and diffusions and between semilinear parabolic equations and superdiffusions. (Diffu- sions and superdiffusions in Part 1 are time inhomogeneous processes.) In Part 2 we deal with elliptic differential equations and with time-homogeneous diffusions and superdiffusions. We apply, when it is possible, the results of Part 1. The most of Part 2 is devoted to the characterization of positive solutions of equation (0.1) by their traces on the boundary and to the study of the boundary singularities of such solutions (both, from analytic and probabilistic points of view). Parabolic counterparts of these results are less complete. Some references to them can be found in bibliographical notes in which we describe the relation of the material presented in each chapter to the literature on the subject. Chapter 1 is an informal introduction where we present some of the basic ideas and tools used in the rest of the book. We consider an elliptic setting and, to simplify the presentation, we restrict ourselves to a particular case of the Laplacian ∆ (for L) and to the Brownian and super-Brownian motions instead of general diffusions and superdiffusions. In the concluding chapter, we give a brief description of some results not in- cluded into the book. In particular, we describe briefly Le Gall’s approach to superprocesses via random snakes (path-valued Markov processes). For a system- atic presentation of this approach we refer to [Le 99a]. We do not touch some other important recent directions in the theory of measure-valued processes: the Fleming-Viot model, interactive measure-valued models... We refer on these sub- jects to Lecture Notes of Dawson [Daw93] and Perkins [Per01]. A wide range of topics is covered (mostly, in an expository form) in “An introduction to Superpro- cesses” by Etheridge [Eth00].
- 13. PREFACE 9 Appendix A and Appendix B contain a survey of basic facts about Markov processes, martingales and elliptic differential equations. A few open problems are suggested in the Epilogue. I am grateful to S. E. Kuznetsov for many discussions which lead to the clarifi- cation of a number of points in the presentation. I am indebted to him for providing me his notes on relations between removable boundary singularities and the Poisson capacity. (They were used in the work on Chapter 13.) I am also indebted to P. J. Fitzsimmons for the notes on his approach to the construction of superprocesses (used in Chapter 4) and to J.-F. Le Gall whose comments helped to fill some gaps in the expository part of the book. I take this opportunity to thank experts on PDEs who gladly advised me on the literature in their field. Especially important was the assistance of N.V. Krylov and V. G. Maz’ya. My foremost thanks go to Yuan-chung Sheu who read carefully the entire man- uscript and suggested numerous corrections and improvements. The research of the author connected with this volume was supported in part by the National Science Foundation Grant DMS -9970942.
- 15. CHAPTER 1 Introduction 1. Brownian and super-Brownian motions and differential equations 1.1. Brownian motion and Laplace equation. Let D be a bounded do- main in Rd with smooth boundary ∂D and let f be a continuous function on ∂D. Then there exists a unique function u of class C2 such that ∆u = 0 in D, u = f on ∂D. (1.1) It is called the solution of the Dirichlet problem for the Laplace equation in D with the boundary value f. A probabilistic approach to this problem can be traced to the classical work [CFL28] of Courant, Friedrichs and Lewy published in 1928. The authors replaced the Laplacian ∆ by its lattice approximation and they represented the solution of the corresponding boundary value problem in terms of the random walk on the lattice. Suppose that a particle starts from a site x in D and moves in one step from a site x to any of 2d nearest neighbor sites with equal probabilities. Let τ be its first exit time from D and ξτ be its location at time τ. Then the solution of the Dirichlet problem on the lattice is given by the formula (1.2) u(x) = Πxf(ξτ ) = Z f(ξτ(ω)(ω))Πx(dω), where Πx is the probability distribution in the path space Ω corresponding to the initial point x. The solution of the problem (1.1) can be obtained by the passage to the limit as the lattice mesh and the duration of each step tend to 0 in a certain relation. In fact, this passage to the limit yields a measure Πx on the space of continuous paths. The stochastic process ξ = (ξt, Πx) is called the Brownian motion and formula (1.2) gives an explicit solution of the problem (1.1) in terms of the Brownian motion ξ. This result is due to Kakutani [Kak44a], [Kak44b]. 1.2. Semilinear equations. Partial differential equations involving a nonlin- ear operator ∆u − ψ(u) appeared in meteorology (Emden, 1897), theory of atomic spectra (Thomas-Fermi, 1920s) and astrophysics (Chandrasekhar, 1937). 1 Since the 1960s, geometers have been interested in these equations in connection with the Yamabe problem: which two functions represent scalar curvature of two Riemannian metrics related by a conformal mapping. The equation (1.3) ∆u = ψ(u) 1See the bibliography in [Vér96]. 11
- 16. 12 1. INTRODUCTION was investigated under various conditions on the function ψ. All these conditions hold for the family (1.4) ψ(u) = uα , α > 1. For a wide class of ψ, the problem ∆u = ψ(u) in D, u = f on ∂D, (1.5) has a unique solution under the same conditions on D and f as the classical problem (1.1). However, analysts discovered a number of new phenomena related to this equation. In 1957 Keller [Kel57a] and Osserman [Oss57] found that all positive solutions of (1.3) are uniformly locally bounded. The most work was devoted to the case of ψ given by (1.4). In 1974, Loewner and Nirenberg [LN74] proved that, in an arbitrary domain D, there exists the maximal solution. This solution tends to ∞ at ∂D if D is bounded and ∂D is smooth. 2 In 1980 Brezis and Véron [BV80] showed that the maximal solution in the punctured space Rd {0} is trivial if d ≥ κα = 2α α − 1 and it is equal to q | x |−2/(α−1) with q = [2(α − 1)−1 (κα − d)]1/(α−1) if d < κα. 1.3. Super-Brownian approach to semilinear equations. A probabilistic formula (1.2) for solving the problem (1.1) involves the value of f at a random point ξτ on the boundary. The problem (1.5) can be approached by introducing, instead, a random measure XD on ∂D and by taking the integral hf, XDi of f with respect to XD. The probability law Pµ of XD depends on an initial measure µ and the role similar to that of (1.2) is played by the formula (1.6) u(x) = − logPxe−hf,XD i . Here Px stands for Pµ corresponding to the initial state µ = δx (unit mass con- centrated at x). We call (XD, Pµ) the exit measure from D. Heuristically, we can think of a random cloud for which XD is the mass distribution on an absorbing barrier placed on ∂D. We consider families of exit measures which we call branching exit Markov (shortly, BEM) systems because their principal characteristics are a branching prop- erty and a Markov property. The BEM system used in formula (1.6) is called the super-Brownian motion. In the next section we explain how it can be obtained by a passage to the limit from discrete BEM systems. Before that, we give, as the first application of (1.6), an expression for a solution exploding on the boundary. Note that, if XD 6= 0, then e−ch1,XDi → 0 as c → +∞ and, if XD = 0, then e−ch1,XDi = 1 for all c. Therefore a solution tending to ∞ at ∂D can be expressed by the formula (1.7) u(x) = − log Px{XD = 0}. 2They considered, in connection with a geometric problem, a special case α = d+2 d−2 .
- 17. 1. BROWNIAN AND SUPER-BROWNIAN MOTIONS AND DIFFERENTIAL EQUATIONS 13 The fact that u is finite is equivalent to the property Px{XD = 0} > 0, i.e., the cloud is extinct in D with positive probability. 1.4. Super-Brownian motion. We start from a system of Brownian parti- cles which die at random times leaving a random number of offspring N with the generating function EzN = ϕ(z). The following picture 3 explains the construction of the exit measure (XD, Pµ). We have here a particle system started by two particles located at points x1, x2 of D. At the death time, the first particle creates two children who survives until they reach ∂D at points y1, y2. Of three children of the second particle, one hits ∂D at point y3, one dies childless and one has two children. Only one child reaches the boundary (at point y4). y4 y1 x2 y3 x1 y2 Figure 1 The initial and exit measures are given by the formulae µ = X δxi XD = X δyi where δc is the unit mass concentrated at c. This way we arrive at a family X of integer-valued random measures (XD, Pµ) where D is an arbitrary bounded open set and µ is an arbitrary integer-valued measure. Since particles do not interact, we have (1.8) Pµe−hf,XD i = e−hu,µi where (1.9) u(x) = − logPxe−hf,XD i . We call this relation the branching property. We also have the following Markov property: for every C ∈ F⊃D and for every µ, (1.10) Pµ{C | F⊂D} = PXD (C) Pµ-a.s. Here F⊂D and F⊃D are the σ-algebras generated by XD0 , D0 ⊂ D and by XD00 , D00 ⊃ D. If the mass of each particle is equal to β, then the initial measure and the exit measures take values 0, β, 2β, . . .. We pass to the limit as β and the expected life time of particles tend to 0 and the initial number of particles tends to infinity. In 3Of course, this is only a scheme. Path of the Brownian motion are very irregular which is not reflected in our picture.
- 18. 14 1. INTRODUCTION the limit, we get an initial measure on D and an exit measure on ∂D which are not discrete. We denote them again µ and XD. The branching property and the Markov property are preserved under this passage to the limit and we get a BEM system (XD, Pµ) where D is an arbitrary bounded open set and µ is an arbitrary finite measure. A function ψ obtained by a passage to the limit from ϕ belongs to a class Ψ0 which contains ψ(u) = uα with 1 < α ≤ 2 but not with α > 2. The probability distribution of the random measure (XD, Pµ) is described by (1.8)–(1.9) and u is a solution of the integral equation (1.11) u(x) + Πx Z τ 0 ψ[u(ξs)]ds = Πxf(ξτ ). If ∂D is smooth and f is continuous, then (1.11) implies (1.5). Hence, (1.6) is a solution of the problem (1.5). Formulae (1.8) and (1.11) determine the probability distribution of XD for a fixed D. Joint probability distributions of XD1 , · · · , XDn can be defined recursively for every n by using the branching and Markov properties. The following equations, similar to (1.8) and (1.11), describe the mass distri- bution Xt at time t: Pµ exph−f, Xti = exph−ut, µi, (1.12) ut(x) + Πx Z t 0 ψ[ut−s(ξs)]ds = Πxf(ξt). (1.13) We cover both sets of equations, (1.8), (1.11) and (1.12), (1.13), by considering exit measures (XQ, Pµ) for open subsets Q of the time-space S = R × Rd and measures µ on S. They satisfy the equations Pµ exph−f, XQi = exph−u,µi, (1.14) u(r, x) + Πr,x Z τ r ψ[u(s, ξs)]ds = Πr,xf(τ, ξτ ) (1.15) where τ = inf{t : (t, ξt) / ∈ Q} is the first exit time from Q. Note that Xt = XS<t where S<t = (−∞, t) × Rd . If Q = (−∞, t) × D where D is a bounded smooth domain 4 and if f is bounded and continuous, then u is a solution of a parabolic equation (1.16) u̇ + 1 2 ∆u = ψ(u) in Q such that u = f on ∂Q. The maximal solution of the equation (1.3) can be described through the range of X. This is the minimal closed set R which contains the support supp Xt for all t. (It contains, a.s., supp XD for each D.) 5 For every open set D, a maximal solution in D is given by (1.17) u(x) = − log Px{R ⊂ D}. 4The name “smooth” is used for domains of class C2,λ (see section 6.1.3). 5Writing “a.s.” means Pµ-a.s. for all µ [or Πµ-a.s. for all µ in the case of a Brownian motion].
- 19. 2. EXCEPTIONAL SETS IN ANALYSIS AND PROBABILITY 15 2. Exceptional sets in analysis and probability 2.1. Capacities. The most important class of exceptional sets in analysis are sets of Lebesgue measure 0. The next important class are sets of capacity 0. A capacity is a function C(B) ≥ 0 defined on all Borel sets. 6 It is not necessarily additive but it is monotone increasing and continuous with respect to the monotone increasing limits. For every B, C(B) is equal to the supremum of C(K) over all compact sets K ⊂ B and it is equal to the infimum of C(O) over all open sets O ⊃ B. [A more systematic presentation of Choquet’s capacities is given in section 10.3.2] To every random closed set (F(ω), P) there corresponds a capacity (2.1) C(B) = P{F ∩ B 6= ∅}. Another remarkable class of capacities correspond to pairs (k, k·k) where k(x, y) is a function on the product space E × Ẽ and k ·k is a norm in a space of functions on E. The most important are the uniform norm (2.2) kfk = sup x |f(x)| and the Lα (m)-norms (2.3) kfkα = Z |f(x)|α m(dx) 1/α where 1 ≤ α ∞ and m is a measure on E. We assume that E and Ẽ are nice metric spaces and that k(x, y) is positive valued, lower semicontinuous in x and measurable in y. To every measure ν on Ẽ there corresponds a function (2.4) Kν(x) = Z Ẽ k(x, y) ν(dy) on E. The capacity corresponding to (k, k · k) is defined on subsets B of Ẽ by the formula (2.5) C(B) = sup{ν(B) : ν is concentrated on B and kKνk ≤ 1}. Our primary interest is not in capacities themselves but rather in the classes of sets on which they vanish, and we say that two capacities are of the same type if these classes coincide. 2.2. Exceptional sets for the Brownian motion. The Brownian motion ξ in a domain D killed at the first exit time τ from D has a transition density pt(x, y). If D is bounded, then (2.6) g(x, y) = Z ∞ 0 pt(x, y)dt is finite for x 6= y. We call g(x, y) Green’s function. The Green’s capacity corre- sponds to the kernel g(x, y) and the uniform norm (2.2). 7 It is of the same type as the capacity corresponding to (2.7) g1(x, y) = Z ∞ 0 e−t pt(x, y) dt 6And even on a larger class of analytic sets 7If d 1, then g(x, x) = ∞. Therefore Green’s capacity of a single point is equal to 0.
- 20. 16 1. INTRODUCTION (and to the uniform norm). In the case d ≥ 3, it is also of the same type as the capacity corresponding to the kernel |x − y|2−d . For a bounded set D, τ ∞ a.s. The range R of ξ is a continuous image of a compact set [0, τ] and therefore, for every x ∈ D, (R, Πx) is a random closed set. Consider the corresponding capacity (2.8) Cx(B) = Πx{R ∩ B 6= ∅}. A set B is called polar for ξ if Cx(B) = 0 for all x ∈ D B. This is equivalent to the condition (2.9) Πx{ξt ∈ B for some t} = 0 for all x ∈ D B [in other words, a.s., ξ does not hit B ]. It is well-known (see, e.g., [Doo84]) that a set B is polar if and only if its Green’s capacity is equal to 0. This gives an analytic characterization of the class of polar sets. 2.3. Exceptional sets for the super-Brownian motion. We say that a set B is polar for X if it is not hit by the range of X, that is if (2.10) Px{R ∩ B 6= ∅} = 0 for all x / ∈ B. In other words, B is polar, if, for all x / ∈ B, Capx (B) = 0 where Capx is the capacity associated with a random closed set (R, Px). It was proved in [Dyn91c] that all capacities Capx are of the same type as the capacity determined by the kernel (2.7) and the norm (2.3) (assuming that ψ is given by (1.4)). It is clear from (1.17) that a closed set B is polar for X if and only if equation (1.3) has only a trivial solution u = 0 in Rd B. By the analytic result described in section 1.2, a single point is polar if and only if d ≥ κα. 2.4. Exceptional boundary sets. Suppose that D is a bounded smooth domain. Denote by γ(dy) the normalized surface area on ∂D. 8 If τ is the first exit time of the Brownian motion ξ from D, then, for every Borel (or analytic) subset Γ of ∂D, (2.11) Πx{ξτ ∈ Γ} = Z Γ k(x, y)γ(dy), x ∈ D where k(x, y) is a strictly positive continuous function on D×∂D called the Poisson kernel. Note that Πx{ξτ ∈ Γ} = 0 if and only if γ(Γ) = 0. In other words, the capacity corresponding to a random closed set ({ξτ }, Πx) is of the same type as the measure γ. A class of exceptional boundary sets related to the super-Brownian motion X is more interesting. It can be defined probabilistically in terms of the range RD of X in D — the minimal closed subset supporting XD0 for all D0 ⊂ D. Or it can be introduced analytically via the capacity CPα corresponding to the Poisson kernel k and the Lα (m)-norm (2.12) kfkα = Z D |f(x)|α m(dx) 1/α . Here m(dx) = dist(x, ∂D)dx. It is proved in Chapter 13 that (2.13) Px{RD ∩ Γ 6= ∅} = 0 for all x ∈ D, 8This is a measure on ∂D determined by the Riemannian metric induced on ∂D by the Euclidean metric in Rd. An explicit expression for γ is given in section 6.1.8.
- 21. 3. POSITIVE SOLUTIONS AND THEIR BOUNDARY TRACES 17 if and only if CPα(Γ) = 0. We call sets Γ with these properties polar boundary sets. The class of such sets can be also characterized by the condition: ν(Γ) = 0 for all ν ∈ N1. Here N1 is a certain set of finite measures on ∂D introduced in Chapter 8. We also establish a close relation between polar boundary sets of the super- Brownian motion and removable boundary singularities for positive solutions of the equation (2.14) ∆u = uα . Namely, we prove that a closed subset Γ of ∂D is polar if and only if it is a removable boundary singularity for (2.14) which means: every positive solution in D equal to 0 on ∂D Γ is identically equal to 0. 3. Positive solutions and their boundary traces 3.1. One of our principal objectives is to describe the class U(D) of all positive solutions of the equation (3.1) ∆u = ψ(u) in an arbitrary domain D. One of the first results in this direction was obtained by Brezis and Véron who proved that, in the case of ψ given by (1.4) and D = Rd {0}, U(D) contains only a trivial solution u = 0 if d ≥ κα (see section 1.2). If 3 ≤ d κα, then U(D) consists of the maximal solution described in section 1.2 and the one- parameter family vc, 0 ≤ c ∞ such that (3.2) vc(x)|x|d−2 → c as x → 0. All positive solutions of the linear equation ∆u = 0 in an arbitrary domain D (that is all positive harmonic functions) were described by Martin. We present a probabilistic version of the Martin boundary theory in Chapter 7. We start the investigation of the class U(D) in Chapter 8 by introducing a subclass U1(D) of moderate solutions which are closely related to harmonic functions. Moderate solu- tions are used as a tool to define, for an arbitrary solution its trace on the boundary. There are two versions of this definition: the rough trace determines a solution only in the case of α (d + 1)/(d − 1). The fine trace is a more complete characteristic. It determines uniquely every σ-moderate solution, that is a solution which is the limit of an increasing sequence of moderate solutions. It remains an open problem if there exist solutions which are not σ-moderate. 3.2. Positive harmonic functions in a bounded smooth domain. We denote by H(D) the class of all positive harmonic functions in a domain D. If D is bounded and smooth, then every h ∈ H(D) has a unique representation (3.3) h(x) = Z ∂D k(x, y)ν(dy) where k is the Poisson kernel and ν is a finite measure on ∂D. We call ν the bound- ary trace of h and we write ν = tr h. Formula (3.3) establishes a 1-1 correspondence between H(D) and the set M(∂D) of all finite measures on ∂D. The constant 1 belongs to H(D) and its trace is the normalized surface area γ (cf. section 2.4). The trace of an arbitrary bounded h ∈ H(D) is a measure
- 22. 18 1. INTRODUCTION absolutely continuous with respect to γ, and the formula (3.4) h(x) = Z ∂D k(x, y)f(y)γ(dy) defines a 1-1 correspondence between bounded h ∈ H(D) and classes of γ-equivalent bounded positive Borel functions on ∂D. It follows from (2.11) that (3.4) is equivalent to (1.2). If f is continuous, then (3.4) is a solution of the Dirichlet problem (1.1). For an arbitrary bounded Borel function f, (3.4) can be considered as a generalized solution of the problem (1.1) because, a.s., h(ξt) → f(ξτ ) as t ↑ τ. An analytic counterpart to this statement is Fatou’s boundary limit theorem: for γ-almost all c ∈ ∂D, h(x) → f(c) as x → c ∈ ∂D non tangentially. It is natural to interpret the measure ν in (3.3) as a weak boundary value of h. In other words, h given by (3.3) can be considered as a solution of a generalized Dirichlet problem ∆h = 0 in D, h = ν on ∂D. (3.5) 3.3. Positive harmonic functions in an arbitrary domain. Martin boundary. Let D be an arbitrary domain and let g(x, y) be given by (2.6). If D is bounded, then g(x, y) ∞ for x 6= y. The same is true for a wide class of unbounded domains. If this is the case, we choose a point c ∈ D and put k(x, y) = g(x, y) g(c, y) . It is possible to imbed D into a compact metric space D̂ = D ∪ Γ and to extend k(x, y) to y ∈ Γ in such a way that yn → y ∈ Γ if and only if k(x, yn) → k(x, y) for all x ∈ D. Set Γ is called the Martin boundary of D. There exists a Borel subset Γ0 of Γ such that every h ∈ H(D) has a unique representation (3.6) h(x) = Z Γ0 k(x, y) ν(dy) where ν ∈ M(Γ0 ). We write ν = tr h and we denote by γ the trace of h = 1. There exists, a.s., a limit of ξt in D̂ as t ↑ τ. It belongs to Γ0 . We denote it ξτ−. The trace of a bounded harmonic function has a form fdγ and, a.s., h(ξt) → f(ξτ−) as t ↑ τ. 3.4. Moderate solutions. We say that a solution u of (3.1) is moderate and we write u ∈ U1(D) if there exists h ∈ H(D) such that u ≤ h. In Chapter 8 we prove that formula (3.7) u(x) + Z D g(x, y)ψ(y) dy = h(x) establishes a 1-1 correspondence between U1(D) and a subclass H1(D) of H(D). Moreover, h is the minimal harmonic function dominating u and u is the maximal element of U(D) dominated by h. The class H1(D) can be characterized by the condition: h ∈ H1(D) if and only if the trace of h does not charge exceptional
- 23. 3. POSITIVE SOLUTIONS AND THEIR BOUNDARY TRACES 19 boundary set (described in section 2.4). If u corresponds to h and if tr h = ν, then u can be considered as a solution of a generalized Dirichlet problem ∆u = ψ(u) in D, u = ν on ∂D (3.8) (cf. (3.5)). It is natural to call ν the boundary trace of a moderate solution u. 3.5. Rough trace. For every u ∈ U(D) and for every closed subset B of ∂D we define the sweeping QB(u) of u to B. In the case of a smooth domain, QB(u) is the maximal solution dominated by u and equal to 0 on ∂D B. [The definition is more complicated in the case of an arbitrary domain.] The rough trace of u is a pair (Γ, ν) where Γ is a closed subset of ∂D and ν is a Radon measure on O = ∂D Γ. Namely, Γ is the minimal closed set such that QB(u) is moderate for all B disjoint from Γ. The measure ν is determined by the condition: the restriction of ν to every B ⊂ O is equal to the trace of the moderate solution QB(u). The main results about the rough trace presented in Chapter 10 are: A. Characterization of all pairs (Γ, ν) which are traces. [The principal condition is that ν(B) = 0 for all exceptional boundary sets.] B. Existence of the maximal solution with a given trace and an explicit prob- abilistic formula for this solution. Le Gall’s example (presented in section 3.5 of Chapter 10) shows that, in gen- eral, infinitely many solutions can have the same rough trace. 3.6. Fine trace. Again this is a pair (Γ, ν) where Γ is a subset of ∂D and ν is a measure on O = ∂D Γ. However Γ is not necessarily closed and ν is not necessarily Radon measure. Roughly speaking, the set Γ consists of points of the boundary near which u rapidly tends to infinity. A precise definition can be formulated, both, in analytic and probabilistic terms. Here we sketch a probabilistic approach based on the concept of a Brownian motion in D conditioned to exit from D at a given point y of the boundary. This stochastic process is described by a measure Πy x on the space of continuous paths which start at point x ∈ D and which are at y at the fist exit time τ from D. Let f be a positive Borel function in D. We say that y is a point of rapid growth of f if, for every x, Z τ 0 f(ξs) ds = ∞ Πy x-a.s. We say that y is a singular point of a solution u if it is a point of rapid growth of function ψ0 (u). We define Γ as the set of all singular points of u. To define the measure ν, we consider all moderate solutions v ≤ u with the trace not charging Γ. ν is the minimal measure such that, for every such v, tr v ≤ ν. We prove that: A. A pair (Γ, ν) is a trace if and only if ν does not charge exceptional boundary sets and if Γ contains all singular points of the following two solutions: u∗ = sup {moderate v with the trace dominated by ν}, uΓ = sup {moderate v with the trace concentrated on Γ}
- 24. 20 1. INTRODUCTION B. Among the solutions with a given trace, there exists a minimal solution and this solution is σ-moderate. 9 C. A σ-moderate solution is determined uniquely by its trace. The solutions in Le Gall’s example are uniquely characterized by their fine traces. 9See the definition in section 3.1.
- 25. Part 1 Parabolic equations and branching exit Markov systems
- 27. CHAPTER 2 Linear parabolic equations and diffusions We introduce diffusions by using analytic results on fundamental solutions of parabolic differential equations. A probabilistic approach to boundary value prob- lems is based on the Perron method in PDEs. A central role is played by Poisson’s and Green’s operators which we define in terms of diffusions. Fundamental concepts of regular boundary points and of regular domains are also defined in probabilistic terms. 1. Fundamental solution of a parabolic equation 1.1. Operator L. We work with functions u(r, x), r ∈ R, x ∈ E = Rd on (d + 1)-dimensional Euclidean space S = R × E. The first coordinate of a point z ∈ S is interpreted as a time parameter. We write u̇ for ∂u ∂r and Diu for ∂u ∂xi where x1, . . ., xd are coordinates of x. Put Dij = DiDj. Operator L is defined by the formula (1.1) Lu(r, x) = d X i,j=1 aij(r, x)Diju(r, x) + d X i=1 bi(r, x)Diu(r, x) where aij = aji. We assume that: 1.1.A. There exists a constant κ 0 such that X aij(r, x)titj ≥ κ X t2 i for all (r, x) ∈ S, t1, . . ., td ∈ R. [ κ is called the ellipticity coefficient of L .] 1.1.B. aij and bi are bounded continuous and satisfy a Hölder’s type condition: there exist constants 0 λ 1 and Λ 0 such that |aij(r, x) − aij(s, y)| ≤ Λ(|x − y|λ + |r − s|λ/2 ), (1.2) |bi(r, x) − bi(r, y)| ≤ Λ|x − y|λ (1.3) for all r, s ∈ R, x, y ∈ E. For every interval I, we denote by SI or S(I) the slab I × E. We write St for SI with I = (−∞, t) and we write Qt for the intersection of Q with St. . Writing U b Q means that Q and U are open subsets of S, U is bounded and its closure Ū is contained in Q. We use the name a sequence exhausting Q for a sequence of open sets Qn ↑ Q such that Qn b Qn+1 for all n. 23
- 28. 24 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS 1.2. Equation u̇ + Lu = 0. We investigate equation 1 (1.4) u̇ + Lu = 0 in Q. Speaking about solutions of (1.4), we assume that the partial derivatives u̇, Diu, i = 1, . . ., d and Diju, i, j = 1, . . ., d are continuous. We denote C2 (Q) the class of functions with this property. Another class of functions plays a special role – continuous functions on Q that are locally Hölder continuous in x uniformly in r. More precisely, we put u ∈ Cλ (Q) if u(r, x) is continuous on Q and if, for every compact Γ ⊂ Q, there exists a constant ΛΓ such that |u(r, x) − u(r, y)| ≤ ΛΓ|x − y|λ for all (r, x), (r, y) ∈ Γ. [λ (called Hölder’s exponent) satisfies the condition 0 λ 1.] 1.3. Fundamental solution. The following results are proved in the theory of partial differential equations (see Chapter 1 in [Fri64] and section 4 in [IKO62]). Theorem 1.1. There exists a unique continuous function p(r, x; t, y) on the set {r t, x, y ∈ E} with the properties: 1.3.A. For every (t, y), the function u(r, x) = p(r, x; t, y) is a solution of (1.5) u̇ + Lu = 0 in St. 1.3.B. For every t1 t2 and every δ 0, the function p(r, x; t, y) is bounded on the set {t1 r t t2, t − r + |y − x| ≥ δ}. 1.3.C. If ϕ is continuous at a and bounded, then Z E p(r, x; t, y)ϕ(y) dy → ϕ(a) as r ↑ t, x → a. Function p is strictly positive and Z E p(r, x; t, y) dy = 1 for all r t and all x; (1.6) Z E p(r, x; s, y)p(s, y; t, z) dy = p(r, x; t, z) for all r s t and all x, z. (1.7) Function p(r, x; t, y) is called a fundamental solution of equation (1.5). We say that a function f is exp-bounded on B if supB |f(r, x)|e−β|x|2 ∞ for every β 0. (Clearly, all bounded functions are exp-bounded.) We use the following properties of a fundamental solution. 1.3.1. If κ is the ellipticity coefficient of L, then, for every β κ, (1.8) p(r, x; t, y) ≤ C(t − r)−d/2 exp −β|y − x|2 2(t − r) for all t1 r t t2, x, y ∈ E where the constant C depends on t1, t2 and β. 1This equation can be reduced by the time reversal r → −r to the equation u̇ = Lu which is usually considered in the literature on partial differential equations. The form (1.4) is more appropriate from a probabilistic point of view.
- 29. 2. DIFFUSIONS 25 1.3.2. If ϕ is an exp-bounded function on St = {t} × E, then (1.9) u(r, x) = Z E p(r, x; t, y)ϕ(y) dy is exp-bounded on S[t0 , t) for every finite interval [t0 , t) and it satisfies equation u̇ + Lu = 0 in St. If, in addition, ϕ is continuous, then, for every t0 t, u is a unique exp-bounded solution of the problem u̇ + Lu = 0 in S(t0 , t), u = ϕ on St. (1.10) [Writing u = ϕ at z̃ ∈ ∂Q means u(z) → ϕ(z̃) as z ∈ Q tends to z̃.] 1.3.3. If ρ is a bounded Borel function on S(t0 , t) and if (1.11) v(r, x) = Z t r ds Z E p(r, x; s, y)ρ(s, y) dy, then Div are continuous on S(t0 , t) [and therefore v ∈ Cλ [S(t0 , t)]]. If, in addition, ρ ∈ Cλ [S(t0 , t)], then v is a unique bounded solution of the problem v̇ + Lv = −ρ in S(t0 , t), v = 0 on St. (1.12) 2. Diffusions 2.1. Continuous strong Markov processes. Here we describe a class of Markov processes which contains all diffusions in a d-dimensional Euclidean space E. 2 Imagine a particle moving at random in E. Suppose that the motion starts at time r at a point x and denote ξt the state at time t ≥ r. The probability that ξt belongs to a set B depends on r and x and we assume that it is equal to R B p(r, x; t, y) dy. Moreover, we assume that, for every n = 1, 2, . . . and for all r t1 · · · tn and all Borel sets B1, . . ., Bn, (2.1) Probability of the event{ξt1 ∈ B1, . . ., ξtn ∈ Bn} = Z B1 dy1 . . . Z Bn dyn p(r, x; t1, y1)p(t1, y1; t2, y2) . . .p(tn−1, yn−1; tn, yn). If the conditions (1.6)-(1.7) are satisfied, then the results of computation with different n do no contradict each other and, by a Kolmogorov’s theorem 3 there exists a probability measure Πr,x on the space of all paths in E starting at time r which satisfies (2.1). We say that p(r, x; t, y) is the transition density of the stochastic process (ξt, Πr,x). Sometimes the measures Πr,x can be defined on the space of all continuous paths. For instance, this is possible for (2.2) p(r, x; t, y) = [2π(t − r)]−d/2 exp − |x − y|2 2(t − r) . The corresponding continuous process is called the Brownian motion . Diffusions also have continuous paths. (Their transition densities will be defined in section 2.2.) 2Basic facts on Markov processes are presented more systematically in the Appendix A. 3See [Kol33], Section III.4. Two proofs of Kolmogorov’s theorem are presented in [Bil95].
- 30. 26 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS We denote by Ωr the space of all continuous paths ω(t), t ∈ [r, ∞) in E. To deal with a single space Ω, we introduce an extra point [ and we put ω(t) = [ for ω ∈ Ωr and t r. We consider ξt as a function on Ω, namely, ξt(ω) = ω(t). The birth time α is a function on Ω equal to r on Ωr. Measure Πr,x is concentrated on the set {α = r, ξα = x}. For every interval I, the σ-algebra F(I) generated by ξs, s ∈ I can be viewed as the class of all events determined by the behavior of the path during I. Note that {α ≤ t} = {ξt ∈ E} belongs to F(I) for all I which contain t. We use an abbreviation F≥t = F[t, ∞). Every process (ξt, Πr,x) satisfies the following condition (which is called the Markov property): events observable before and after time t are conditionally inde- pendent given ξt. More precisely, if r t, A ∈ F[r, t] and B ∈ F≥t, then (2.3) Πr,x(AB) = Z A Πt,ξt (B)Πr,x(dω). To simplify notation we write z for (r, x) and ηt for (t, ξt). Formula (2.3) implies that for all X ∈ F[r, t] and every Y ∈ F≥t, 4 (2.4) Πz(XY ) = Πz(XΠηt Y ). Diffusions satisfy a stronger condition called the strong Markov property. Roughly speaking, it means that (2.4) can be extended to all stopping times τ. The defi- nition of stopping times and their properties are discussed in the Appendix A. An important class of stopping times are the first exit times. The first exit time from an open set Q is defined by the formula (2.5) τ(Q) = inf{t ≥ α : ηt / ∈ Q}. [We put τ(Q) = ∞ if ηt ∈ Q for all t ≥ α.]We say that X is a pre-τ random variable if X1τ≤t ∈ F≤t for all t. In the Appendix A we give a general formulation of the strong Markov property, we prove it for a wide class of Markov processes which includes all diffusions and we deduce from it propositions 2.1.A–2.1.C — the only implications which we need in this book. 2.1.A. Let ρ be a positive Borel function on S. For every stopping time τ and every pre-τ X ≥ 0, (2.6) ΠzX Z ∞ τ ρ(ηs) ds = ΠzXGρ(ητ ) where (2.7) Gρ(z) = Πz Z ∞ α ρ(ηs) ds. 2.1.B. Let τ0 be the first exit time from an open set Q0 . Then for every stopping time τ ≤ τ0 , for every pre-τ X ≥ 0 and for every Borel function f ≥ 0, (2.8) ΠzX1Q0 (ητ )1τ0∞f(ητ0 ) = ΠzX1τ∞1Q0 (ητ )KQ0 f(ητ ) where (2.9) KQ0 f(z) = Πz1τ0∞f(ητ0 ). 4Writing X ∈ F means that X ≥ 0 and X is measurable with respect to a σ-algebra F. It can be proved (by using Theorem 1.1 in the Appendix A) that every X ∈ F≤t coincide Πr,x -a.e. with a F[r, t]-measurable function. Therefore (2.4) holds for all X ∈ F≤t.
- 31. 2. DIFFUSIONS 27 [The value of η∞ is not defined. Instead of introducing in (2.8) and (2.9) factors 1τ0∞ and 1τ∞, we can agree to put f(η∞) = 0.] 2.1.C. Suppose that V is an open subset of S × S and τ is a stopping time. Put (2.10) σt = inf{u ≥ α : u t, (ηt, ηu) / ∈ V }. If στ ∞ Πz-a.s. for all z, then, for every pre-τ function X ≥ 0 and every Borel function f ≥ 0, (2.11) ΠzXf(ηστ ) = ΠzXF(ητ ) where (2.12) F(t, y) = Πt,yf(ησt ). 2.2. L-diffusion. An L-diffusion is a continuous strong Markov process with transition density p(r, x; t, y) which is a fundamental solution of (1.5). The existence of such a process is proved in Chapter 5 of [Dyn65]. Note that (2.13) Πr,xϕ(ξt) = Z E p(r, x; t, y)ϕ(y) dy. It follows from Fubini’s theorem that (2.14) Πr,x Z t r ρ(s, ξs) ds = Z t r ds Z E p(r, x; s, y)ρ(s, y) dy. Therefore, under the conditions on ϕ and ρ formulated in 1.3.2–1.3.3, (2.15) u(r, x) = Πr,xϕ(ξt) is a solution of the problem (1.10) and (2.16) v(r, x) = Πr,x Z t r ρ(s, ξs) ds is a solution of the problem (1.12). 2.3. Martingales associated with L-diffusions. Martingales are one of new tools contributed to analysis by probability theory. 5 The following theorem establishes a link between martingales and parabolic differential equations. Theorem 2.1. Suppose f ∈ C2 (S(t1, t2)) is exp-bounded on S(t1, t2) and that ρ = ˙ f + Lf is bounded and belongs to Cλ (S(t1, t2)). Then, for every r ∈ (t1, t2) and every x ∈ E, (2.17) Yt = f(ηt) − Z t r ρ(ηs) ds, t ∈ (r, t2) is a martingale with respect to F[r, t] and Πr,x. Proof. First, we prove that, for all r t and all x, (2.18) w(r, x) = Πr,x[f(ηt) − f(ηr) − Z t r ρ(ηs) ds] is equal to 0. Indeed, w = u−f −v where u is defined by (2.15) with ϕ(x) = f(t, x) and v is defined by (2.16). It follows from 1.3.2–1.3.3, that w is an exp-bounded 5See the Appendix A for basic facts on martingales.
- 32. 28 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS solution of problem (1.10) with ϕ = 0. Such a solution is unique and therefore w = 0. For every t, Yt is measurable relative to F[r, t] and Πr,x|Yt| ∞. We need to prove that, for all r ≤ t0 t and for every bounded F[r, t0 ]-measurable X, (2.19) Πr,xX(Yt − Yt0 ) = 0. Note that Yt − Yt0 = f(ηt) − f(ηt0 ) − Z t t0 ρ(ηs) ds is F≥t0-measurable. By the Markov property (2.4), Πr,xX(Yt − Yt0 ) = Πr,xXΠt0,ξt0 (Yt − Yt0 ) and (2.19) holds because, by (2.18), Πt0,y(Yt − Yt0 ) = 0 for all y. Corollary. Suppose that U b Q and let τ be the first exit time from U. If f ∈ C2 (Q) and if ρ = ˙ f + Lf ∈ Cλ (Q), then (2.20) Πr,xf(ητ ) = f(r, x) + Πr,x Z τ r ( ˙ f + Lf)(ηs) ds. Proof. Since U is bounded, it is contained in SI for some finite interval I. There exists a bounded function of class C2 (SI) which coincides with f on Ū and therefore we can assume that f is defined on SI and that it satisfies the conditions of Theorem 2.1. The martingale Yt given by (2.17) is continuous and τ is bounded Πr,x-a.s. By Theorem 4.1 in the Appendix A, Πr,xYτ = Πr,xYr which implies (2.20). 3. Poisson operators and parabolic functions 3.1. Poisson operators. The Poisson operator corresponding to an open set Q is defined by the formula (3.1) KQf(z) = Πz1τ∞f(ητ ) where τ is the first exit time from Q (cf. formula (2.9)). Note that KQf = f on Qc . It follows from 2.2.1.B that, for every U b Q and every f ≥ 0, (3.2) KU KQf = KQf. 3.2. Parabolic functions. We say that a continuous function u in Q is par- abolic if, for every open set U b Q, (3.3) KU u = u in U. The following lemma is an immediate implication of Corollary to Theorem 2.1. Lemma 3.1. Every solution u of the equation (3.4) u̇ + Lu = 0 in Q is a parabolic function in Q. We say that a Borel subset T of ∂Q is total if, for all z ∈ Q, Πz{τ ∞, ητ ∈ T } = 1. In particular, ∂Q is total if and only if Πz{τ = ∞} = 0 for all z ∈ Q. [This condition holds, for instance, if Q ⊂ St with a finite t.] If ∂Q is not total, then there exist no total subsets of ∂Q.
- 33. 3. POISSON OPERATORS AND PARABOLIC FUNCTIONS 29 Lemma 3.2. Suppose T is a total subset of ∂Q. If u is bounded and continuous on Q ∪ T and if it is parabolic in Q, then (3.5) KQu = u in Q. Proof. Consider a sequence Qn exhausting Q. The sequence τn = τ(Qn) is monotone increasing. Denote its limit by σ. For almost all ω, σ ≤ τ ∞ and ησ ∈ T . Therefore σ = τ(Q). We get (3.5) by passing to the limit in the equation u(z) = Πzu(ητn ). Lemma 3.3. Suppose that parabolic functions un converge to u at every point of Q. If un are locally uniformly bounded, then u is also parabolic. Proof. If U b Q, then un are uniformly bounded on Ū. By passing to the limit in the equation KU un = un, we get KU u = u. 3.3. Poisson operator corresponding to a cell. Subsets of S of the form C = (a0, b0) × (a1, b1) × · · · × (ad, bd) are called (open) cells. Points of ∂C with the first coordinate equal to a0 form the bottom B of C. Clearly, T = ∂C B is a total subset. We denote it ∂rC. A basic result proved in every book on parabolic equations 6 implies that, if f is a bounded continuous function on ∂rC, then there exists a continuous function u on C ∪ ∂rC such that u̇ + Lu = 0 in C, u =f on ∂rC. (3.6) It follows from Lemmas 3.1 and 3.2 that u = KCf. Note that KC is continuous with respect to the bounded convergence. It follows from the multiplicative systems theorem (Theorem 1.1 in the Appendix A) that these two properties characterize KC. This provides a purely analytic definition of KC. A particular class of cells is defined by the formula C(z, β) = {z0 : d̃(z, z0 ) β} where d̃(z, z0 ) = maxi |xi − x0 i| for z = (x0, . . ., xd), z0 = (x0 0, . . ., x0 d). 3.4. Superparabolic and subparabolic functions. A lower semicontinu- ous function u is called superparabolic if, for every open set U b Q, (3.7) KU u ≤ u in U. A function u is called subparabolic if −u is superparabolic. Lemma 3.4. Suppose that u is a bounded below lower semicontinuous function in Q and that (3.7) holds for every cell C b Q. Then u is superparabolic and, moreover, (3.7) holds for all U ⊂ Q. Proof. For U = S, the relation (3.7) is satisfied because its left side is 0. If U 6= S, then d(z) = inf z0∈∂U d̃(z, z0 ) ∞ for all z. Put V = {(z, z̃) : ˜ d(z, z̃) 1 2 d(z)} 6See, e.g., Chapter 3, section 4 in [Fri64] or Chapter V, section 2 in [Lie96].
- 34. 30 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS and consider the function σt defined by the formula (2.10). The stopping times τ0 = α, τn+1 = στn for n ≥ 0 are finite and τ0 ≤ τ1 ≤ · · · ≤ τn ≤ . . .. It follows from 2.1.C that Πzu(ητn+1 ) = ΠzF(ητn ) where F(z) = Πzu(ητ1 ). If (3.7) is satisfied for cells, then F(z) ≤ u(z). Hence, Πzu(ητn+1 ) ≤ Πzu(ητn ) and, by induction, Πzu(ητn ) ≤ Πzu(ητ0 ) = u(z). We have d̄(ητn+1 , ητn) = d(ητn )/2. If τ is the limit of τn, then, on the set {τ ∞}, ητn → ητ and therefore 0 = d̃(ητ , ητ ) = d(ητ )/2. We conclude that {τ ∞} ⊂ {ητ ∈ ∂U} ⊂ {τ = τ(U)}. By the definition of the lower semicontinuity, on the set {τ ∞}, u(ητ ) ≤ lim inf u(ητn ). Therefore, by Fatou’s lemma, Πz1τ∞u(ητ ) ≤ Πz1τ∞ lim inf u(ητn ) ≤ lim inf Πz1τ∞u(ητn ) ≤ u(z). Lemma 3.5. Suppose that w is superparabolic in Q and bounded below. Let T be a total subset of ∂Q. If, for every z̃ ∈ T , (3.8) lim inf w(z) ≥ 0 as z → z̃, then w ≥ 0 in Q. Proof. Let τ = τ(Q). It follows from Lemma 3.4 that Πz1τ∞w(ητ ) ≤ w(z). Condition (3.8) implies that w(ητ ) ≥ 0 Πz-a.s. on {τ ∞}. Hence w(z) ≥ 0. 3.5. The Perron solution. The following analytic results are proved, for instance, in [Lie96], Chapter III, section 4. 7 Let f be a bounded Borel function on ∂Q. A bounded below superparabolic function w is in the upper Perron class U+ for f, if (3.9) lim inf z→z̃ w(z) ≥ f(z̃) for all z̃ ∈ ∂Q. Analogously, a bounded above subparabolic function v is in the lower Perron class U− for f, if lim sup z→z̃ w(z) ≤ f(z̃) for all z̃ ∈ ∂Q. It follows from Lemma 3.5 that v ≤ w for every v ∈ U− and every w ∈ U+. Since f is bounded, all sufficiently big constants are in U+ and all sufficiently small constants are in U−. Therefore functions of class U− are uniformly bounded from above and functions of class U+ are uniformly bounded from below. It is proved that the infimum u of all functions w ∈ U+ coincides with the supremum of all functions v ∈ U−. Moreover, u is a solution of the equation (3.4). It is called the Perron solution corresponding to f. Theorem 3.1. If Q is a bounded open set and f is a bounded Borel function on ∂Q, then u = KQf is the Perron solution corresponding to f. Proof. Let w ∈ U+ and let τ be the first exit time from Q. Consider a sequence Qn exhausting Q and the corresponding first exit times τn. We have w(z) ≥ Πzw(ητn ) and, by condition (3.9) and Fatou’s lemma, w(z) ≥ lim inf Πzw(ητn ) ≥ Πz lim inf w(ητn ) ≥ Πzf(ητ ). Similarly, if v ∈ U−, then v(z) ≤ Πzf(ητ ). 7See also [Doo84], 1.XVIII.1. There only the case L = ∆ is considered but the arguments can be modified to cover a general L.
- 35. 3. POISSON OPERATORS AND PARABOLIC FUNCTIONS 31 Corollary. A function u is parabolic in Q if and only if it is a solution of (3.4). 3.6. Smooth superparabolic functions. The improved Maximum prin- ciple. We wish to prove: 3.6.A. If u ∈ C2 (Q) and if (3.10) u̇ + Lu ≤ 0 in Q, then U is superparabolic in Q. This follows immediately from (2.20) if u̇ + Lu ∈ Cλ (Q). To eliminate this restriction, we use: 3.6.B. Suppose that C is a cell and u ∈ C2 (C) satisfies the conditions u̇ + Lu ≥ 0 in C, lim supu(z) ≤ 0 as z → z̃ for all z̃ ∈ ∂rC. Then u ≤ 0 in C. [This proposition is proved in any book on parabolic PDEs (for instance, in Chapter 2 of [Fri64] or in Chapter II of [Lie96]).] To prove 3.6.A we consider an arbitrary cell C b Q. As we know, v = KCu is a solution of the problem (3.6) with f equal to the restriction of u to ∂rC. Therefore w = v − u satisfies conditions ẇ + Lw ≥ 0 in C and w(z) → 0 as z → z̃ ∈ ∂rC. By 3.6.B, w ≤ 0 in C. Hence, KCu ≤ u. By Lemma 3.4, u is superparabolic. 3.6.C. [The improved maximum principle.] Let T be a total subset of ∂Q. If v ∈ C2 (Q) is bounded above and satisfies the condition (3.11) v̇ + Lv ≥ 0 in Q and if, for every z̃ ∈ T , (3.12) lim supv(z) ≤ 0 as z → z̃, then v ≤ 0 in Q. Indeed, by 3.6.A, u = −v is superparabolic and, by Lemma 3.5, u ≥ 0. 3.7. Superparabolic functions and supermartingales. Proposition 3.1. Suppose u is a positive lower semicontinuous superparabolic function in Q and τ = τ(Q). Then, for every r, x, Xt = 1tτ u(ηt) is a supermartingale on [r, ∞) relative to F[r, t] and Πr,x. Proof. Note that σ = τ ∧ t is the first exit time from Q ∩ St. Since σ ∞, by Lemma 3.4, for every s t, Πs,xu(ησ) ≤ u(s, x). Since {σ τ} = {σ = t}, we have Πs,xXt = Πs,x1σ=tu(ηt) = Πs,x1σ=tu(ησ) ≤ u(s, x).
- 36. 32 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS Let τr be the first after r exit time from Q. If r s, then {τr s} ∈ F[r, s]. Clearly, τr = τ Πr,x-a.s. If A ∈ F[r, s], then {A, s τr} ∈ F[r, s] and, by the Markov property (2.4), Z A,sτr Xt dΠr,x = Z A,sτr Πηs Xt dΠr,x ≤ Z A,sτr u(ηs) dΠr,x. For s t, Xt1sτr = Xt Πr,x-a.s. and therefore R A Xt dΠr,x ≤ R A Xs dΠr,x. Since Xt is F[r, t]-measurable and Πr,x-integrable, it is a supermartingale. 4. Regular part of the boundary 4.1. Regular points. A point z̃ = (r̃, x̃) of ∂Q is called regular if, for every t r̃, (4.1) Πz̃{ηs ∈ Q for all s ∈ (r̃, t)} = 0. . Theorem 4.1. Let τ be the first exit time from Q. If a point z̃ = (r̃, x̃) ∈ ∂Q is regular then, for every t r̃, (4.2) Πz{τ t} → 0 as z ∈ Q tends to z̃. Proof. 1◦ . Fix t and put, for every r ≤ s t, A(s, t) = {ηu ∈ Q for u ∈ (s, t)} and qr s(x) = Πr,xA(s, t). Note that qr r(x) = Πr,x{τ t} for (r, x) ∈ Q. Therefore the conditions (4.1) and (4.2) are equivalent to the conditions qr̃ r̃(x̃) = 0 and qr r(x) → 0 as (r, x) → (r̃, x̃). 2◦ . By the Markov property of ξ, for all r ≤ s t, qr s(x) = Πr,xΠs,ξs A(s, t) = Z E p(r, x; s, y)qs s(y) dy. 3◦ . It follows from 2◦ and 1.3.2 that qr s(x) is continuous in (r, x) for r s. Therefore, for every ε 0, there exists a neighborhood U of (r̃, x̃) such that |qr s(x) − qr̃ s(x̃)| ε for all (r, x) ∈ U. 4◦ . Clearly, qr s(x) ↓ qr r (x) as s ↓ r. 5◦ . Suppose r̃ t. If (r̃, x̃) is regular, then, by 1◦ , qr̃ r̃(x̃) = 0 and, by 4◦ , for every ε 0, there exists s ∈ (r̃, t) such that qr̃ s(x̃) ε. By 3◦ , if (r, x) ∈ U, then qr r(x) ≤ qr s(x) ≤ qr̃ s(x̃) + |qr s(x) − qr̃ s(x̃)| 2ε. By 1◦ , this implies (4.2). Remark. The converse to Theorem 4.1 is also true: (4.2) implies (4.1). We do not use this fact. In an elliptic setting, it is proved in Chapter 13 of [Dyn65]. The role of condition (4.2) is highlighted by the following theorem: Theorem 4.2. If (4.2) holds at z̃ ∈ ∂Q and if a bounded function f on ∂Q is continuous at z̃, then (4.3) KQf(z) → f(z̃) as z → z̃.
- 37. 4. REGULAR PART OF THE BOUNDARY 33 Informally, we have the following implications: (4.4) {z = (r, x) ∈ Q is close to z̃ = (r̃, x̃)} =⇒ {τ is close to r} =⇒ {ητ is close to z and therefore close to z̃} =⇒ {f(ητ ) is close to f(z̃)} =⇒ {Πzf(ητ ) is close to f(z̃)}. A rigorous proof is based on the following lemma. Lemma 4.1. Fix t ∈ R and put (4.5) Dr = sup rst |ηs − ηr|. For every ε 0, there exits δ 0 such that (4.6) Πr,x{Dr ε} ε for all x and all r ∈ (t − δ, t). Proof. If ξs = (ξ1 s , . . ., ξd s ), then Dr ≤ t − r + d X 1 Di r where Di r = sup rst |ξi s − ξi r|. To prove the lemma it is sufficient to show that, for every ε 0, there exists δ 0 such that (4.7) Πr,x{Di r ε} ε for all x and all r ∈ (t − δ, t). Each function fi(r, x) = xi is exp-bounded on S and ρi = ˙ fi + Lfi = bi (a coefficient in (1.1)). The conditions of Theorem 2.1 hold for fi on every interval (t1, t2). Therefore, for every r ∈ (t1, t2), Y i s = ξi s − Z s r ρi(ηu) du, s ∈ (r, t2) is a martingale relative to F[r, s], Πr,x. Choose (t1, t2) which contains [r, t]. By Kolmogorov’s inequality (see section 4.4 in the Appendix A) (4.8) Πr,x{ sup rst |Y i s − Y i r | δ} ≤ δ−2 Πr,x|Y i t − Y i r |2 . If |ρi| ≤ c, then (4.9) Di r ≤ sup rst |Y i s − Y i r | + c(t − r). On the other hand, since (A + B)2 ≤ 2A2 + 2B2 for all A, B, we have (4.10) Πr,x|Y i t − Y i r |2 ≤ 2Πr,x|ξi t − ξi r|2 + 2c2 (t − r)2 . The bound (1.8) implies that (4.11) Πr,x(ξi t − ξi r)2 = Z E p(r, x; t, y)(yi − xi)2 dy ≤ Z E p(r, x; t, y)|y − x|2 dy ≤ C(t − r) where C is a constant [depending on t1, t2]. The bound (4.7) follows from (4.8)– (4.11).
- 38. 34 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS Proof of Theorem 4.2. Let z = (r, x), z̃ = (r̃, x̃). For every t r̃, Πr,x{|ητ − z| ε} ≤ Πr,x{τ ≥ t} + Πr,x{Dr ε}. By Lemma 4.1, for every ε 0, there exists δ 0 such that Πr,x{Dr ≥ ε} ε for all x and all r ∈ (t − δ, t). Choose t ∈ (r̃, r̃ + δ/2). Then every r ∈ (r̃ − δ/2, r̃) belongs to (t − δ, t). By Theorem 4.1, Πr,x{τ ≥ t} ε in a neighborhood U of z̃. Suppose that r ∈ (t − δ, t) and z ∈ U. Then Πr,x{|ητ − z| ε} ≤ 2ε. Let V be the intersection of U with the ε-neighborhood of z̃. If z ∈ V , then Πr,x{|ητ − z̃| 2ε} ≤ Πr,x{|ητ − z| ε} ≤ 2ε. Suppose that N is an upper bound for |f| and let |f(z)−f(z̃)| ε for z ∈ V . Then Πr,x|f(ητ ) − f(z̃)| ≤ 2NΠr,x{|ητ − z̃| δ} + ε ≤ (2N + 1)ε in V which implies (4.3). Theorem 4.3. Suppose Q is bounded and all points of a total subset T of ∂Q are regular. If a function f is bounded and continuous on T , then u = KQf is a unique bounded solution of the problem u̇ + Lu = 0 in Q, u = f on T . (4.12) Proof. Put f = 0 on ∂QT . By Theorems 3.1 and 4.2, u = KQf is a solution of the problem (4.12). Clearly, u is bounded. For an arbitrary bounded solution v of (4.12), by Lemmas 3.1 and 3.2, v = KQv = KQf = u. We say that a function u is a barrier at z̃ if there exists a neighborhood U of z̃ such that u ∈ C2 (U) ∩ C(Ū) and (4.13) u̇ + Lu ≤ 0 in Q ∩ U, u(z̃) = 0, u 0 on Ū ∩ Q̄ except z̃. Lemma 4.2. The condition (4.2) holds if there exists a barrier u at z̃. 8 Proof. Put V = Q ∩ U. For every t r̃, the infimum β of u on the set ∂V ∩ S≥t is strictly positive. Let τ = τ(V ). By Chebyshev’s inequality and (3.1), (4.14) Πz{τ t} ≤ Πz{u(ητ ) ≥ β} ≤ Πzu(ητ )/β = KV u(z)/β. By 3.6.A, u is superparabolic in V . Denote by f the restriction of u to ∂V . Clearly, u belongs to the upper Perron class for f. By Theorem 3.1, KV f is the correspond- ing Perron solution. Hence, KV u = KV f ≤ u, and (4.14) implies (4.2). By constructing a suitable barrier, we prove that (4.2) holds if z̃ can be touched from outside by a ball. More precisely, we have the following test. 8The existence of a barrier is also a necessary condition for the regularity of z̃. (See, e.g., [Lie96], Lemma 3.23 or [Dyn65], Theorem 13.6.)
- 39. 4. REGULAR PART OF THE BOUNDARY 35 Theorem 4.4. The property (4.2) holds at z̃ = (r̃, x̃) ∈ ∂Q if there exists z0 = (r0 , x0 ) with x0 6= x̃ such that |z − z0 | |z̃ − z0 | for all z ∈ Q̄ sufficiently close to z̃ and different from z̃. In other words, z̃ is the only common point of three sets: Q̄, a closed ball centered at z0 and a neighborhood of z̃. Proof. We claim that, if ε = |z̃ − z0 | and if p is sufficiently large, then u(z) = ε−2p − |z − z0 |−2p , is a barrier at z̃. Clearly, u(z̃) = 0. There exists a neighborhood U of z̃ such that, for all z ∈ Ū ∩ Q̄, |z − z0 | ε and therefore u(z) 0. We have u̇ + Lu = A[−(p + 1)B + C] where A = 2p|z − z0 |−2(p+2) , B = X aij(xi − x0 i)(xj − x0 j), C = |z − z0 |2 X [aii + bi(xi − x0 i)]. Note that B ≥ κ|x − x0 |2 where κ is the ellipticity coefficient of L. Since aij and bi are bounded, we see that, for sufficiently large p, u̇ + Lu ≤ 0 in a neighborhood of z̃ assuming that x̃ 6= x0 . 4.2. Regular open sets. We denote by ∂regQ the set of all regular points of ∂Q and by ∂rQ the set of all interior (relative to ∂Q) points of ∂regQ. We say that Q is regular if ∂regQ contains a total subset of ∂Q. A smaller class of strongly regular open sets is defined by the condition: ∂rQ is total in ∂Q. For a cell C, ∂rC coincides with the set introduced in section 3.3. This set is relatively open in ∂C and therefore cells are strongly regular open sets. Note that the following conditions are equivalent: (a) B is a relatively open subset of A; (b) B = A ∩ O where O is an open subset of S; (c) A = B ∪ F where F is a closed subset of S. Therefore, if Bi is a relatively open subset of Ai, i = 1, 2, then B1 ∩B2 is relatively open in A1 ∩A2 and B1 ∪B2 is relatively open in A1 ∪A2. Lemma 4.3. If U is strongly regular, then Q = U ∩ Q1 is strongly regular for every open set Q1 such that Ū ∩ ∂Q1 ⊂ ∂rQ1. Proof. The boundary ∂Q is the union of three sets A1 = ∂U ∩ Q1, A2 = U ∩ ∂Q1 and A3 = ∂U ∩∂Q1. Sets B1 = ∂rU ∩Q1, B2 = U ∩∂rQ1 and B3 = ∂rU ∩∂rQ1 are relatively open in, respectively, A1, A2, A3 and therefore T = B1 ∪ B2 ∪ B3 is relatively open in ∂Q. Every point of T is regular in ∂Q. It remains to show that T is total in ∂Q. Let τ be the first exit time from Q and let z ∈ Q. Since ∂rU is total in ∂U and ∂rU ∩ (A1 ∪ A3) ⊂ B1 ∪ B3, we have {ητ ∈ A1 ∪ A3} ⊂ {ητ ∈ B1 ∪ B3} Πz-a.s. On the other hand, A2 ⊂ ∂rQ1 and therefore A2 = B2 and {ητ ∈ A2} = {ητ ∈ B2}. Now we introduce an important class of simple open sets. We start from closed cells [a0, b0] × [a1, b1] × · · · × [ad, bd]. We call finite unions of closed cells simple compact sets. We define a simple open set as the collection of all interior points of a simple compact set . The boundary ∂C of a cell C = [a0, b0]×[a1, b1]×· · ·×[ad, bd] consists of 2(d + 1) d-dimensional faces. We distinguish two horizontal faces: the top {b0} × [a0, b0] × [a1, b1] × · · ·× [ad, bd] and the bottom {a0} × [a0, b0] × [a1, b1] × · · · × [ad, bd]. We call the rest side faces.
- 40. 36 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS Theorem 4.5. Every simple open set is strongly regular. For an arbitrary open set Q, there exists a sequence of simple open sets exhausting Q. In the proof we use the following observations. 4.2.A. Let H be a (d − 1)-dimensional affine subspace of Rd [that is the set of x = (x1, . . ., xd) such that a1x1 + · · · + adxd = c for some constants a1, . . ., ad not all equal to 0]. Then for all r t, x ∈ Rd , Πr,x{ξt ∈ H} = 0. If H is a (d − 2)-dimensional affine subspace, then Πr,x{ξt ∈ H for some t r} = 0. The first part holds because the probability distribution of ξt is absolutely continuous with respect to the Lebesgue measure. We leave the second part as an exercise for a reader. 4.2.B. If F is a (d − 1)-dimensional face of a cell C, then Πr,x{(t, ξt) ∈ F for some t r} = 0 for all (r, x) ∈ S. This follows easily from 4.2.A. Proof of Theorem 4.5. Every compact simple set A can be represented as the union of closed cells C1, . . ., Cn such that the intersection of every two distinct cells Ci, Cj is either empty or it is a common face of both cells. Let Q be the set of all interior points of A. Note that ∂Q = ∪N 1 Fk where F1, . . ., FN are d-dimensional cells which enter the boundary of exactly one of Ci. Clearly, the set F0 k of all points of Fk that do not belong to any (d − 1)-dimensional face of any Ci is open in ∂Q. By 4.2.B, to prove that Q is strongly regular, it is sufficient to show that, for every k, either Πz{ητ ∈ F0 k } = 0 for all z ∈ Q or F0 k ⊂ ∂regQ. Clearly, the first case takes place if Fk is the bottom of Ci. If Fk is the top of Ci, then, obviously, F0 k ⊂ ∂regQ. If Fk is a side face, then F0 k ⊂ ∂regQ by Theorem 4.4. It remains to construct sets Qn. It is easy to reduce the general case to the case of a bounded Q. Suppose that Q is bounded. Put εn = (d + 1)1/2 2−n . Consider a partition of S = Rd+1 into cells with vertices in the lattice 2−n Zd+1 and take the union An of all cells whose εn-neighborhood are contained in Q. The set Qn of all interior points of An is a simple open set. Clearly, the sequence Qn exhaust Q. For every two sets A, B, we denote by d(A, B) the infimum of d(a, b) = |a − b| over all a ∈ A, b ∈ B. Suppose that Q is an open set and Γ is a closed subset of ∂Q. We say that a sequence of open sets Qn ↑ Q is a (Q, Γ)-sequence if Qn are bounded and strongly regular and if (4.15) Q̄n ↑ Q̄ Γ; d(Qn, Q Qn+1) 0. Lemma 4.4. A (Q, Γ)-sequence exists if Γ contains all irregular points of ∂Q. Proof. By Theorem 4.5, there exists a sequence of strongly regular open sets Un exhausting S Γ. If Γ contains all irregular points of ∂Q, then, by Lemma 4.3, sets Qn = Un ∩ Q are strongly regular. Note that Q̄n ⊂ Q̄ and Q̄n ∩ Γ ⊂ Ūn ∩ Γ ⊂ Un+1 ∩ Γ = ∅. Hence Q̄n ⊂ Q̄ Γ. If K is a compact set disjoint from Γ, then K ⊂ Un for some n. Let x ∈ Q̄ Γ. For sufficiently small δ 0, K = {y : |y −x| ≤ δ} is disjoint from Γ. If xm → x and xm ∈ Q, then, for sufficiently large m, xm ∈ Un ∩ Q = Qn. Hence x ∈ Q̄n. This proves the first part of (4.15). The second part holds because Qn ⊂ Ūn, QQn+1 ⊂ Uc n+1 and d(Ūn, Uc n+1) 0.
- 41. 5. GREEN’S OPERATORS AND EQUATION u̇ + Lu = −ρ 37 5. Green’s operators and equation u̇ + Lu = −ρ 5.1. Parts of a diffusion. A part ˜ ξ of a diffusion ξ in an arbitrary open set Q ⊂ S is obtained by killing ξ at the first exit time τ from Q. More precisely, we consider ˜ ξt = ξt for t ∈ [α, τ), = † for t ≥ τ where † – “the cemetery” – is an extra point (not in E). The state space at time t is the t-section Qt = {x : (t, x) ∈ Q} of Q. We will show that ˜ ξ = (˜ ξt, Πr,x) is a Markov process with the transition density (5.1) pQ(r, x; t, y) = p(r, x; t, y) − Πr,xp(τ, ξτ ; t, y) for x ∈ Qr, y ∈ Qt. [We set p(r, x; t, y) = 0 for r ≥ t.] Theorem 5.1. For every Borel function f ≥ 0 on Q, (5.2) Πr,x1tτ f(t, ξt) = Z Qt pQ(r, x; t, y)f(t, y) dy for all x ∈ Qr. Moreover, for every n = 1, 2, . . . and for all r t1 · · · tn, x ∈ Qr and Borel sets B1, . . ., Bn, (5.3) Πr,x{τ tn, ξt1 ∈ B1, . . ., ξtn ∈ Bn} = Z B1 dy1 . . . Z Bn dynpQ(r, x; t1, y1)pQ(t1, y1; t2, y2) . . .pQ(tn−1, yn−1; tn, yn). We have: pQ(r, x; t, y) ≥ 0 for all r t, x ∈ Qr, y ∈ Qt; (5.4) Z Qt pQ(r, x; t, y) dy ≤ 1 for all r t and all x ∈ Qr; (5.5) (5.6) Z Qs pQ(r, x; s, y)pQ(s, y; t, z) dy = pQ(r, x; t, z) for all r s t and all x ∈ Qr, z ∈ Qt. Proof. If we set f = 0 outside Q, then (5.7) Πr,x1τ=tf(t, ξt) = 0 because f(τ, ξτ ) = 0. Therefore (5.8) Πr,x1tτ f(t, ξt) = u(r, x) − v(r, x) where u(r, x) = Πr,xf(t, ξt), v(r, x) = Πr,x1τtf(t, ξt). By (1.9) and (2.15), (5.9) u(s, x) = Z E p(s, x; t, y)f(t, y)dy. By 2.1.B (applied to τ0 = t), (5.10) v(r, x) = Πr,x1τtf(t, ξt) = Πr,x1τtF(τ, ξτ )
- 42. 38 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS where F(s, w) = Πs,wf(t, ξt) = Z E p(s, w; t, y)f(t, y) dy. Formula (5.2) follows from (5.8), (5.9), (5.10) and (5.1). We establish (5.3) by induction by applying (5.2) and the Markov property (2.4). To prove (5.4), (5.5) and (5.6), we establish that pQ(r, x; t, y) is continuous in y ∈ Qt for every r t and every x ∈ Qr. This follows from a similar property of p(r, x; t, y) because (τ, ξτ ) ∈ ∂Q Πr,x-a.s. and, by 1.3.B, p(τ, ξτ ; t, y) is uniformly bounded in a neighborhood of each y ∈ Qt. Formula (5.2) implies (5.4) and (5.5). To prove (5.6), we note that, for r s t, Πr,x{τ t, ξs ∈ Qs, ξt ∈ B} = Πr,x{τ t, ξt ∈ B} By (5.3), this implies that the functions of z in both parts of (5.6) have the same integrals over B. Therefore (5.6) holds for almost all z. It holds for all z because both parts are continuous in z. Formula (5.3) is an analog of formula (2.1) for a process with a random death time τ. 5.2. Green’s functions. We prove that function pQ defined by (5.1) has properties similar to 1.3.A–1.3.C. We call it Green’s function for operator u̇ + Lu in Q. 5.2.A. If Qt = St ∩Q is bounded, then for every (t, y) ∈ Q, function u(r, x) = pQ(r, x; t, y) is a solution of (3.4) in Qt. Indeed, for every (t, y) ∈ Q, u = ũ−KQũ where ũ(r, x) = p(r, x; t, y). By 1.3.A, ũ satisfies (3.4) in St. By 1.3.B, ũ is bounded on ∂Q and, by Theorem 3.1 KQũ satisfies (3.4) in Qt. 5.2.B. For every t1 t2 and every δ 0, function pQ(r, x; t, y) is bounded on the intersection of Q with the set {t1 r t t2, t − r + |y − x| ≥ δ}. This follows from 1.3.B because pQ ≤ p. 5.2.C. If ϕ is bounded and continuous at a ∈ Qt, then Z Qt pQ(r, x; t, y)ϕ(y) dy → ϕ(a) as (r, x) → (t, a), (r, x) ∈ Qt. Proof. By (5.1) and (5.10), 0 ≤ Z Qt (p − pQ)(r, x; t, y)ϕ(y) dy = Πr,x Z Qt p(τ, ξτ ; t, y)ϕ(y) dy = Πr,x1τtϕ(ξt), and 5.2.C follows from 1.3.C if we prove that (5.11) u(r, x) = Πr,x{τ t} → 0 as (r, x) → (t, a), (r, x) ∈ Qt. Note that u = KQt 1St . If a ∈ Qt, then (t, a) is a regular point of ∂Qt. Since 1St is continuous and equal to 0 at (t, a), formula (5.11) follows from Theorem 4.2.
- 43. 5. GREEN’S OPERATORS AND EQUATION u̇ + Lu = −ρ 39 There exists a simple relation between pQ and pQ0 for Q0 ⊂ Q: (5.12) pQ0 (r, x; t, y) = pQ(r, x; t, y) − Πr,xpQ(τ0 , ξτ0 ; t, y) where τ0 = τ(Q0 ). Indeed, put f(r, x) = p(r, x; t, y) and denote by fQ, fQ0 the functions obtained in a similar way from pQ, pQ0 . By (5.1), fQ = f − KQf and, by (3.2), KQ0 fQ = KQ0 f − KQf. Hence, fQ − fQ0 = −KQf + KQ0 f = KQ0 fQ. 5.3. Green’s operators. Green’s operator in an arbitrary open set Q is de- fined by the formula (5.13) GQρ(r, x) = Πr,x Z τ r ρ(s, ξs) ds (cf. (2.7)). By (5.2), (5.14) GQρ(r, x) = Z ∞ r ds Z Qs pQ(r, x; s, y)ρ(s, y)dy. If Q0 ⊂ Q, then, by (5.14) and (5.12), (5.15) GQ = GQ0 + KQ0 GQ. 5.3.A. Suppose that Q ⊂ SI = I × E, I is a finite interval and ρ is bounded. Then function w = GQρ belongs to Cλ (Q). If ρ ∈ Cλ (Q), then w ∈ C2 (Q) and it is a solution of the equation (5.16) ẇ + Lw = −ρ in Q. If ρ is bounded and if z̃ is a regular point of ∂Q, then (5.17) w(z) → 0 as z → z̃. Proof. Note that w = v − KQv where v is given by (1.11). Since KQv is parabolic in Q, the first part of 5.3.A follows from 1.3.3. If z = (r, x) and if N is an upper bound of |ρ|, then, for every ε 0, |w(z)| ≤ N[(t − r)Πz{τ r + ε} + ε] and therefore (5.17) follows from Theorem 4.1. 5.3.B. Let τ be the first exit time from an arbitrary open set Q. If ρ ≥ 0 and w = GQρ is finite at a point z ∈ Q, then (5.18) lim t↑τ w(ηt) = 0 Πz-a.s. Proof. Let z = (r, x). We can assume that ρ ≥ 0. We prove that Mt = 1tτ w(ηt), t ∈ [r, ∞) is a supermartingale relative to F[r, t], Πr,x. To this end, we consider a bounded positive F[r, t]-measurable function X and we note that, by the Markov property (2.4), Πr,xX1tτ Z τ t ρ(s, ξs) ds = Πr,xX1tτ Πt,ξt Z τ t ρ(s, ξs) ds = Πr,xX1tτ w(t, ξt). Hence Πr,xXMt ≤ Πr,xXMs for r ≤ s ≤ t. Since Mt is F[r, t]-measurable and Πr,x-integrable, our claim is proved. Since Mt is right continuous, a limit Mτ− as t ↑ τ exists Πr,x-a.s. (see 4.3.C in the Appendix A). Suppose Qn exhaust Q and let τn be the first exit time from Qn.
- 44. 40 2. LINEAR PARABOLIC EQUATIONS AND DIFFUSIONS By (5.15), w = GQn ρ + KQn w. Since GQn ρ ↑ GQρ, we conclude that KQn w ↓ 0 and Πr,xMτ− = Πr,x limw(τn, ξτn ) ≤ limΠr,xw(τn, ξτn ) = limKQn w = 0. 5.3.C. If Q, ρ and f are bounded and if f is continuous on ∂regQ, then (5.19) v = GQρ + KQf is a solution of the problem v̇ + Lv = −ρ in Q, v = f on ∂regQ. (5.20) This follows from 5.3.A and Theorems 3.1 and 4.2. 5.3.D. Let ρ be bounded. A function w is a solution of equation (5.16) if and only if w is locally bounded and, for every U b Q, (5.21) w = GU ρ + KU w. Proof. Suppose that w satisfies (5.16). By Theorem 4.5, for an arbitrary U there exists a sequence of regular open sets Un ↑ U. Since w is bounded and continuous on Ū, we have KUn w → KU w. Also GUn ρ → GUρ. Therefore it is sufficient to prove (5.21) for a regular U. By 5.3.A and Theorem 4.3, u = GUρ + KU w − w is a solution of the problem u̇ + Lu = 0 in U, u = 0 on ∂regU. (5.22) By 3.6.C, u = 0. If w satisfies (5.21) and is bounded on Ū, then the equation (5.16) holds on U by 5.3.A and Theorem 3.1. 5.3.E. Suppose that solutions wn of (5.16) converge to w at every point of Q. If wn are locally uniformly bounded, then w also satisfies (5.16). This follows from 5.3.D (cf. the proof of Lemma 3.3). 6. Notes Our treatment of diffusions is in spirit of the book [Dyn65]. However, in this book only time-homogeneous case was considered. Inhomogeneous diffusions were covered in [Dyn93]. In particular, one can found there a probabilistic formula for the Perron solutions, the improved maximum principle and an approximation of arbitrary domains by simple domains. A concept of strongly regular domains was introduced in [Dyn98a]. This class of domains plays a special role in the theory of semilinear partial differential equations (see, Chapter 5). A fundamental monograph of Doob [Doo84] contains the most complete pre- sentation of the connections between the Brownian motion and classical potential theory related to the Laplace equation. Bibliographical notes in [Doo84] should be consulted for the early history of this subject. A special role in the book is played by martingale theory. Much of this theory was created by Doob. Martingale are the principal tool used by Stroock and Varadhan to develop a new approach to diffusions. A construction of diffusions by solving a martingale problem is presented in their monograph [SV79].
- 45. 6. NOTES 41 A direct construction of the paths of diffusions by solving stochastic differential equations is due to Itô [Itô51]. A modern presentation of Itô’s calculus and its applications is given in the books of Ikeda and Watanabe [IW81] and Rogers and Williams [RW87].
- 47. CHAPTER 3 Branching exit Markov systems In this chapter we introduce a general model — BEM systems — which is the basis for the theory of superprocesses and, in particular, superdiffusions to be developed in the next chapters. A BEM system describes a mass distribution of a random cloud started from a distribution µ and frozen at the exit from Q. Mathematically, this is a family X of random measures (XQ, Pµ) in a space S. The parameter Q takes values in a class of subsets of S, µ is a measure on S, XQ is a function of ω ∈ Ω and Pµ is a probability measure on Ω. A Markov property is defined with the role of “past” and “future” played by Q0 ⊂ Q and Q00 ⊃ Q. [This definition can be applied to a parameter Q taking values in any partially ordered set.] We consider systems which combine the Markov property with a branching property which means, heuristically, an absence of interaction between any parts of the random cloud described by X. We start from historical roots of the concept of branching. Then we introduce branching particle systems (they were described on a heuristic level in Chapter 1) and we use them to motivate a general definition of BEM systems. The transition operators VQ play a role similar to the role of the transition functions in the theory of Markov processes. We investigate properties of these operators and we show how a BEM system can be constructed starting from a family of operators VQ. At the end of the chapter some basic properties of BEM systems are proved. 1. Introduction 1.1. Simple models of branching. The first probabilistic model of branch- ing appeared in 1874 in the problem of the family name extinction posed by Francis Galton and solved by H. W. Watson [WG74]. Galton’s motivation was to evaluate a conjecture that the extinction of prominent families is more likely than the ex- tinction of ordinary ones. He suggested to start from probabilities pn for a man to have n sons evaluated by the demographical data for the general population. The problem consisted in computation of the probability of extinction after k genera- tions. Watson’s solution contained an error but he introduced a tool of fundamental importance for the theory of branching. The principal observation was: if ϕ(z) = ∞ X 0 pnzk is the generating function for the number of sons, then the generating function ϕk for the number of descendants in the k-th generation can be evaluated by the recursive formula (1.1) ϕk+1 = ϕ(ϕk). 43
- 48. 44 3. BRANCHING EXIT MARKOV SYSTEMS The Galton-Watson model and its modifications found many applications in biology, physics, chemistry... 1 A model of branching with a continuous time parameter was suggested in 1947 in [KD47] (an output of a Kolmogorov’s seminar held at Moscow University in 1946-47). Consider a particle system and assume that a single particle produces, during time interval (r, t), k = 0, 1, 2, . . . particles with probability pk(r, t). Gener- ating functions ϕ(r, t; z) = ∞ X 0 pk(r, t)zk satisfy the condition (1.2) ϕ(r, t; z) = ϕ(r, s; ϕ(s, t; z)) for r s t. Suppose that pk(r − h, r) = ak(r)h + o(h) for k 6= 1, p1(r − h, r) = 1 + a1(r)h + o(h) as h ↓ 0 and let (1.3) Φ(r; z) = ∞ X 0 ak(r)zk . We arrive at a differential equation (1.4) ∂ϕ(r, t; z) ∂r + Φ(r, ϕ(r, t; z)) = 0 for r t with a boundary condition (1.5) Φ(r, t; z) → z as r ↑ t. Equation (1.4) and a linear equation u̇ + Lu = 0 considered in Chapter 2 are particular cases of a semilinear parabolic equation (1.6) u̇ + Lu = ψ(u). A probabilistic approach to (1.6) is based on a model which involves both L-diffusion and branching. 1.2. Exit systems associated with branching particle systems. Con- sider a system of particles moving in E according to the following rules: (1) The motion of each particle is described by a right continuous strong Markov process ξ. (2) A particle dies during time interval (t, t + h) with probability kh + o(h), independently on its age. (3) If a particle dies at time t at point x, then it produces n new particles with probability pn(t, x). (4) The only interaction between the particles is that the birth time and place of offspring coincide with the death time and place of their parent. [Assumption (2) implies that the life time of every particle has an exponential probability distribution with the mean value 1/k.] We denote by Pr,x the probability law corresponding to a process started at time r by a single particle located at point x. Suppose that particles stop to move 1More on an early history of the branching processes can be found in [Har63].
- 49. 1. INTRODUCTION 45 and to procreate outside an open subset Q of S. In other words, we observe each particle at the first, in the family history, 2 exit time from Q. The exit measure from Q is defined by the formula XQ = δ(t1,y1) + · · · + δ(tn,yn) where (t1, y1), . . ., (tn, yn) are the states of frozen particles and δ(t,y) means the unit measure concentrated at (t, y). We also consider a process started by a finite or infinite sequence of particles that “immigrate” at times ri at points xi. There is no interaction between their descendants and therefore the corresponding probability law is the convolution of Pri,xi . We denote it Pµ where µ = X δ(ri,xi) is a measure on S describing the immigration. We arrive at a family X of random measures (XQ, Pµ), Q ∈ O, µ ∈ M where O is a class of open subsets of S and M is the class of all integer-valued measures on S. Family X is a special case of a branching exit Markov system. A general definition of such systems is given in the next section. 1.3. Branching exit Markov systems. A random measure on a measurable space (S, BS) is a pair (X, P) where X(ω, B) is a kernel 3 from an auxiliary mea- surable space (Ω, F) to (S, BS ) and P is a probability measure on F. We assume that S is a Borel subset of a compact metric space and BS is the class of all Borel subsets of S. Suppose that: (i) O is a subset of σ-algebra BS; (ii) M is a class of measures on (S, BS ) which contains all measures δy, y ∈ S. (iii) to every Q ∈ O and every µ ∈ M, there corresponds a random measure (XQ, Pµ) on (S, BS). Condition (ii) is satisfied, for instance, for the class M(S) of all finite measures and for the class N(S) of all integer-valued measures. We use notation hf, µi for the integral of f with respect to a measure µ. Denote by Z the class of functions (1.7) Z = exp{− n X 1 hfi, XQi i} where Qi ∈ O and fi are positive measurable functions on S. We say that X = (XQ, Pµ), Q ∈ O, µ ∈ M is a branching system if 1.3.A. For every Z ∈ Z and every µ ∈ M, (1.8) PµZ = e−hu,µi where (1.9) u(y) = − log PyZ and Py = Pδy . 2By the family history we mean the path of a particle and all its ancestors. If the family history starts at (r, x), then the probability law of this path is Πr,x . 3A kernel from a measurable space (E1, B1) to a measurable space (E2, B2) is a function K(x, B) such that K(x, ·) is a measure on B2 for every x ∈ E1 and K(·, B) is an B1-measurable function for every B ∈ B2.
- 50. 46 3. BRANCHING EXIT MARKOV SYSTEMS Condition 1.3.A (we call it the continuous branching property) implies that PµZ = Y Pµn Z for all Z ∈ Z if µn, n = 1, 2, . . . and µ = P µn belong to M. A family X is called an exit system if: 1.3.B. For all µ ∈ M and Q ∈ O, Pµ{XQ(Q) = 0} = 1. 1.3.C. If µ ∈ M and µ(Q) = 0, then Pµ{XQ = µ} = 1. Finally, we say that X is a branching exit Markov [BEM] system, if XQ ∈ M for all Q ∈ O and if, in addition to 1.3.A–1.3.C, we have: 1.3.D. [Markov property.] Suppose that Y ≥ 0 is measurable with respect to the σ-algebra F⊂Q generated by XQ0 , Q0 ⊂ Q and Z ≥ 0 is measurable with respect to the σ-algebra F⊃Q generated by XQ00 , Q00 ⊃ Q. Then (1.10) Pµ(Y Z) = Pµ(Y PXQ Z). It follows from the principles (1)-(4) stated at the beginning of section 1.2 that conditions 1.3.A–1.3.D hold for the systems of random measures associated with branching particle systems. For them S = R × E, M = N(S) and O is a class of open subsets of S. 1.4. Transition operators. Let X = (XQ, Pµ), Q ∈ O, µ ∈ M be a family of random measures. Denote by B the set of all bounded positive BS-measurable functions. Operators VQ, Q ∈ O acting on B are called the transition operators of X if, for every µ ∈ M and every Q ∈ O, (1.11) Pµe−hf,XQ i = e−hVQ(f),µi . If X is a branching system, then (1.11) follows from the formula (1.12) VQ(f)(y) = − logPye−hf,XQ i for f ∈ B. In this chapter we establish sufficient conditions for operators VQ to be transi- tion operators of a branching exit Markov system. In the next chapter we study a special class of BEM systems which we call superprocesses. A link between operators VQ and a BEM system X is provided by a family of transition operators of higher order VQ1,...,Qn . We call it a V-family. 2. Transition operators and V-families 2.1. Transition operators of higher order. Suppose that (2.1) Pµ exp[−hf1, XQ1 i − · · · − hfn, XQn i], = exp[−hVQ1,...,Qn (f1, . . ., fn), µi]
- 51. 2. TRANSITION OPERATORS AND V-FAMILIES 47 for all µ ∈ M, f1, . . ., fn ∈ B and Q1, . . ., Qn ∈ O. Then we say that operators VQ1,...,Qn are the transition operators of order n for X. Condition (2.1) is equivalent to the assumption that X is a branching system and that (2.2) VQ1,...,Qn (f1, . . ., fn)(y) = − log Py exp[−hf1, XQ1 i − · · · − hfn, XQn i], f1, . . ., fn ∈ B, y ∈ S. [For n = 1, formulae (2.1)–(2.2) coincide with (1.11)–(1.12).] We use the following abbreviations. For every finite subset I = {Q1, . . ., Qn} of O, we put XI = {XQ1 , . . ., XQn },fI = {f1, . . ., fn}, hfI , XIi = n X i=1 hfi, XQi i. (2.3) In this notation, formulae (2.2) and (2.1) can be written as (2.4) VI(fI )(y) = − logPye−hfI ,XIi and (2.5) Pµe−hfI ,XI i = e−hVI (fI ),µi . If X satisfies condition 1.3.C, then: 2.1.A. For every Qi ∈ I, VI(fI ) = fi +VIi (fIi ) on Qc i where Ii is the set obtained from I by dropping Qi. Indeed, hfI , XIi = hfi, XQi i + hfIi , XIi i and hfi, XQi i = fi(y) Py-a.s. if y ∈ Qc i . For a branching exit system X, the Markov property 1.3.D is equivalent to: 2.1.B. If Q ⊂ Qi for all Qi ∈ I, then (2.6) VQVI = VI. Formula (2.6) can be rewritten in the form (2.7) VI(fI) = VQ[1Qc VI (fI)] for all fI. Proof. It follows from (2.5) that e−hVQVI(fI ),µi = Pµe−hVI (fI ),XQi = PµPXQ e−hfI ,XI i . If Q ⊂ Qi for all Qi ∈ I, then hfI , XIi ∈ F⊃Q and 1.3.D implies that the right side is equal to Pµe−hfI ,XI i = e−hVI (fI ),µi . Hence (2.6) follows from 1.3.D. By 1.3.B, for every F, the value of VQ(F) does not depend on the values of F on Q. Therefore (2.7) and (2.6) are equivalent. To deduce 1.3.D from 2.1.B, it is sufficient to prove (1.10) for Y = e−hfI ,XI i , Z = e−hfĨ ,XĨ i where I = {Q1, . . ., Qn}, ˜ I = {Q̃1, . . ., Q̃m} with Qi ⊂ Q ⊂ Q̃j. Note that Y Z ∈ Z. By 1.3.A, the same is true for Y PXQ Z. Therefore (1.10) will follow from 1.3.A if
- 52. 48 3. BRANCHING EXIT MARKOV SYSTEMS we check that it holds for all µ = δy. We use the induction in n. The condition (2.6) implies (2.8) PµZ = PµPXQ Z. Hence, (1.10) holds for n = 0. Suppose it holds for n−1. If y ∈ Qc i , then, by 1.3.C, Py{Y = e−fi(y) Yi} = 1 where Yi = e−hfIi ,XIi i , and we have PyY Z = e−fi(y) PyYiZ = e−fi(y) Py(YiPXQ Z) = Py(Y PXQ Z) by the induction hypothesis. Hence (1.10) holds for δy with y not in the intersection QI of Qi ∈ I. For an arbitrary y, by (2.8), PyY Z = PyPXQI Y Z. By 1.3.B, XQI is concentrated, Py-a.s. on Qc I and therefore PXQI Y Z = PXQI (Y PXQ Z). We conclude that PyY Z = PyPXQI (Y PXQ Z) = Py(Y PXQ Z). Transition operators of order n for a BEM system can be expressed through transition operators of order n − 1 by the formulae (2.9) VI(fI ) = fi + VIi (fIi ) on Qc i for every Qi ∈ I, (2.10) VI (fI) = VQI [1Qc I VI(fI )] where QI is the intersection of all Qi ∈ I. Formula (2.9) (equivalent to 2.1.A) defines the values of VI(fI ) on Qc I. Formula (2.10) follows from (2.7). It provides an expression for all values of VI(fI ) through its values on Qc I. Conditions (2.9)–(2.10) can be rewritten in the form (2.11) VI = VQI ṼI where (2.12) ṼI(fI ) = ( fi + VIi (fIi ) on Qc i , 0 on QI. 2.2. Properties of VQ. We need the following simple lemma. Lemma 2.1. Let Y be a positive random variable and let 0 ≤ c ≤ ∞. If Pe−λY ≤ e−λc for all λ 0, then P{Y ≥ c} = 1. If, in addition Pe−Y = e−c , then P{Y = c} = 1. Proof. If c = ∞, then, P-a.s., e−λY = 0 and therefore Y = ∞. If c ∞, then Pe−λ(Y −c) ≤ 1 and, by Fatou’s lemma, P{ lim λ→∞ e−λ(Y −c) } ≤ 1. Hence, P{Y ≥ c} = 1. The second part of the lemma follows from the first one. Theorem 2.1. Transition operators of an arbitrary system of finite random measures X satisfy the condition: 2.2.A. For all Q ∈ O, (2.13) VQ(fn) → 0 as fn ↓ 0. A branching system X is a branching exit system if and only if:
- 53. 3. FROM A V-FAMILY TO A BEM SYSTEM 49 2.2.B. VQ(f) = VQ( ˜ f) if f = ˜ f on Qc . 2.2.C. For every Q ∈ O and every f ∈ B, VQ(f) = f on Qc . Proof. 1◦ . Property 2.2.A is obvious. It is clear that 1.3.B implies 2.2.B and 1.3.C implies 2.2.C. 2◦ . If 2.2.B holds, then VQ(1Q) = VQ(0) = 0 and therefore Pye−XQ(Q) = 1 which implies 1.3.B. 3◦ . It follows from 2.2.C and (1.11) that, if µ(Q) = 0, then, for all f ∈ B and all λ 0, Pµe−λhf,XQ i = e−λhf,µi and, by Lemma 2.1, (2.14) hf, XQi = hf, µi Pµ-a.s. Since there exists a countable family of f ∈ B which separate measures, 1.3.C follows from (2.14). 2.3. V-families. We call a collection of operators VI a V-family if it satisfies conditions (2.9)–(2.10) [equivalent to (2.11)–(2.12)] and 2.2.A. We say that a V- family and a system of random measures correspond to each other if they are connected by formula (2.1). Theorem 2.2. Suppose that operators VQ, Q ∈ O satisfy conditions 2.2.A– 2.2.C and the condition 2.3.A. For all Q ⊂ Q̃ ∈ O, VQVQ̃ = VQ̃. Then there exists a V-family {VI} such that VI = VQ for I = {Q}. Proof. Denote by |I| the cardinality of I. For |I| = 1, operators VI are defined. Suppose that VI, subject to conditions (2.9)–(2.10), are already defined for |I| n. For |I| = n, we define VI by (2.9)–(2.10). This is not contradictory because fi + VIi (fIi ) = fj + VIj (fIj ) = fi + fj + VIij (fIij ) on Qc i ∩ Qc j. By 2.2.B it is legitimate to define VI (fI) on QI by (2.10). 3. From a V-family to a BEM system 3.1. P-matrices and N-matrices. First, we prepare some algebraic and an- alytic tools. Suppose that a symmetric n×n matrix (aij) satisfies the condition: for all real numbers t1, . . ., tn, n X i,j=1 aijtitj ≥ 0. In algebra, such matrices are called positive semidefinite. Some authors (e.g., [BJR84]) use the name positive definite. We resolve this controversy by using a short name a P-matrix.
- 54. 50 3. BRANCHING EXIT MARKOV SYSTEMS We need another class of matrices which are called negative definite in [BJR84]. [This is inconsistent with the common usage in algebra where negative definite means (−1)× positive definite.] We prefer again a short name. We call an n × n symmetric matrix an N-matrix if n X i,j=1 aijtitj ≤ 0 for every n ≥ 2 and all t1, . . ., tn ∈ R such that P ti = 0. The following property of these classes is obvious: 3.1.A. The classes P and N are closed under entry wise convergence. Moreover, they are convex cones in the following sense: if (B, B, η) is a measure space, if aij(b) is a P-matrix (N-matrix) for all b ∈ B and if aij(b) are η-integrable, then aij = Z aij(b)η(db) is also a P-matrix (respectively, an N-matrix). Here are some algebraic properties of both classes. (i) A matrix (aij) is a P-matrix if and only if it has a representation aij = m X k=1 qikqjk where m ≤ n. This follows from the fact that a quadratic form is positive semidefinite if and only if it can be transformed by a linear transformation to the sum of m ≤ n squares. (ii) If (aij) and (bij) are P-matrices, then so is the matrix cij = aijbij. Indeed, by using (i), we get X ij cijtitj = X k X ij bij(qikti)(qjktj) ≥ 0. (iii) If (aij) is a P-matrix, then cij = eaij is also a P-matrix. This follows from (ii) and 3.1.A. (iv) Suppose that a (n + 1) × (n + 1) matrix (aij)n 0 and an n × n-matrix (bij)n 1 are connected by the formula (3.1) bij = −aij + ai0 + a0j − a00, i, j = 1, . . ., n. Then, for all t0, . . ., tn such that t0 + · · · + tn = 0, n X i,j=0 aijtitj = − n X i,j=1 bijtitj. Therefore (aij) is an N-matrix if and only if (bij) is a P-matrix. Now we can prove the following proposition: 3.1.B. A matrix (aij) belongs to class N if and only if cij(λ) = e−λaij is a P-matrix for all λ 0.
- 55. Exploring the Variety of Random Documents with Different Content
- 56. family necessaries cannot spare the tenth, it may be too much (else even the receivers must all be givers): but when family necessities can spare much more than the tenth, then the tenth is not enough. 3. In those places where church, and state, and poor are all to be maintained by free gift, there the tenth of our increase is far too little, for those that have any thing considerable to spare, to give to all these uses. This is apparent in that the tenths alone were not thought enough even in the time of the law, to give towards the public worship of God: for besides the tenths, there were the first-fruits, and oblations, and many sorts of sacrifices; and yet at the same time, the poor were to be maintained by liberal gifts besides the tenths: and though we read not of much given to the maintenance of their rulers and magistrates, before they chose to have a king, yet afterwards we read of much; and before, the charges of wars and public works lay upon all. In most places with us, the public ministry is maintained by glebe and tithes, which are none of the people's gifts at all, for he that sold or leased them their lands, did suppose that tithes were to be paid out of it, and therefore they paid a tenth part less for it, in purchase, fines, or rents, than otherwise they should have done; so that I reckon, that most of them give little or nothing to the minister at all. Therefore they may the better give so much the more to the needy, and to other charitable uses. But where minister, and poor, and all are maintained by the people's contribution, there the tenths are too little for the whole work; but yet to most, or very many, the tenths to the poor alone, besides the maintenance of the ministry and state, may possibly be more than they are able to give. The tenths even among the heathens, were given in many places to their sacrifices, priests, and to religious, public, civil works, besides all their private charity to the poor. I find in Diog. Laertius, lib. i. (mihi) 32. that Pisistratus the Athenian tyrant, proving to Solon (in his epistle to him) that he had nothing against God or man to blame him for, but for taking the
- 57. crown; telling him, that he caused them to keep the same laws which Solon gave them, and that better than the popular government could have done, doth instance thus: Atheniensium singuli decimas frugum suarum separant, non in usus nostros consumendas, verum sacrificiis publicis, commodisque communibus, et si quando bellum contra nos ingruerit, in sumptus deputandas: that is, Every one of the Athenians do separate the tithes of their fruits, not to be consumed to our uses, but to defray the charge in public sacrifices, and in the common profits, and if war at any time invade us. And Plautus saith, Ut decimam solveret Herculi. Indeed as among the heathens the tithes were conjunctly given for religious and civil uses, so it seems that at first the christian emperors settled them on the bishops for the use of the poor, as well as for the ministers, and church service, and utensils. For to all these they were to be divided, and the bishop was as the guardian of the poor: and the glebe or farms that were given to the church, were all employed to the same uses; and the canons required that the tithes should be thus disposed of by the clergy; non tanquam propriæ, sed domino oblatæ: and the emperor Justinian commanded the bishops, Ne ea quæ ecclesiis relicta sunt sibi adscribant sed in necessarios ecclesiæ usus impendant; lib. xliii. cap. de Episc. et Cler. vid. Albert. Ranzt. Metrop. lib. i. cap. 2. et sax. lib. vi. cap. 52. And Hierom (ad Damasc.) saith, Quoniam quicquid habent clerici pauperum est; et domus illorum omnibus debent esse communes; susceptioni peregrinarum et hospitum invigilare debent; maxime curandum est illis, ut de decimis et oblationibus, cœnobiis et Xenodochiis qualem voluerint et potuerint sustentationem impendant. Yet then the paying of tithes did not excuse the people from all other charity to the poor: Austin saith, Qui sibi aut præmium comparat, aut peccatorum desiderat indulgentiam promereri, reddat decimam, etiam de novem partibus studeat eleemosynam dare pauperibus. And in our times there is less reason that tithes should excuse the people from their works of charity, both because the tithes are now more appropriate to the maintenance of the clergy, and because (as is aforesaid) the people give them not out of their
- 58. own. I confess, if we consider how decimation was used before the law by Abraham and Jacob, and established by the law unto the Jews, and how commonly it was used among the gentiles, and last of all by the church of Christ, it will make a considerate man imagine, that as there is still a divine direction for one day in seven, as a necessary proportion of time to be ordinarily consecrated to God, besides what we can spare from our other days; so that there is something of a divine canon, or direction, for the tenth of our revenues or increase to be ordinarily consecrated to God, besides what may be spared from the rest. And whether those tithes, that are none of your own, and cost you nothing, be now to be reckoned to private men, as any of their tenths, which they themselves should give, I leave to your consideration. Amongst Augustine's works we find an opinion that the devils were the tenth part of the angels, and that man is now to be the tenth order among the angels, the saints filling up the place that the devils fell from, and there being nine orders of angels to be above us, and that in this there is some ground of our paying tenths; and therefore he saith, that Hæc est Domini justissima consuetudo; ut si tu illi decimam non dederis, tu ad decimam revocaberis, id est, dæmonibus, qui sunt decima pars angelorum, associaberis. Though I know not whence he had this opinion, it seemeth that the devoting of a tenth part ordinarily to God, is a matter that we have more than a human direction for. 15. In times of extraordinary necessities of the church, or state, or poor, there must be extraordinary bounty in our contributions: as if an enemy be ready to invade the land, or if some extraordinary work of God (as the conversion of some heathen nations) do require it, or some extraordinary persecution and distress befall the pastors, or in a year of famine, plague, or war, when the necessities of the poor are extraordinary; the tenths in such cases will not suffice, from those that have more to give: therefore in such a time, the primitive christians sold their possessions, and laid down the price at the feet of the apostles. In one word, an honest, charitable heart being presupposed as the root or fountain, and prudence being the discerner of our duty,
- 59. the apostle's general rule may much satisfy a christian for the proportion, 1 Cor. xvi. 2, Let every one of you lay by him in store, as God hath prospered him; and 2 Cor. viii. 12, according to that a man hath: though there be many intimations, that ordinarily a tenth part at least is requisite. III. Having thus resolved the question of the quota pars or proportion to be given, I shall say a little to the question, Whether a man should give most in his lifetime, or at his death? Answ. 1. It is certain that the best work is that which is like to do most good. 2. But to make it best to us, it is necessary that we do it with the most self-denying, holy, charitable mind. 3. That, cæteris paribus, all things else being equal, the present doing of a good work, is better than to defer it. 4. That to do good only when you die, because then you can keep your wealth no longer, and because then it costeth you nothing to part with it, and because then you hope that this shall serve instead of true repentance and godliness; this is but to deceive yourselves, and will do nothing to save your souls, though it do never so much good to others. 5. That he that sinfully neglecteth in his lifetime to do good, if he do it at his death, from true repentance and conversion, it is then accepted of God; though the sin of his delay must be lamented. 6. That he that delayeth it till death, not out of any selfishness, backwardness, or unwillingness, but that the work may be better, and do more good, doth better than if he hastened a lesser good. As if a man have a desire to set up a free-school for perpetuity, and the money which he hath is not sufficient; if he stay till his death, that so the improvement of the money may increase it, and make it enough for his intended work, that is to do a greater good with greater self-denial: for, (1.) He receiveth none of the increase of the money for himself.
- 60. (2.) And he receiveth in his lifetime none of the praise or thanks of the work. So also, if a man that hath no children, have so much land only as will maintain him, and desireth to give it all to charitable uses when he dieth, this delay is not at all to be blamed, because he could not sooner give it; and if it be not in vain-glory, but in love to God and to good works that he leaveth it, it is truly acceptable at last. So that all good works that are done at death, are not therefore to be undervalued, nor are they rejected of God; but sometimes it falleth out that they are so much the greater and better works, though he that can do the same in his lifetime, ought to do it. IV. But though I have spent all these words in answering these questions, I am fully satisfied that it is very few that are kept from doing good by any such doubt or difficulty, in the case which stalls their judgments; but by the power of sin and want of grace, which leave an unwillingness and backwardness on their hearts. Could we tell how to remove the impediments in men's wills, it would do more than the clearest resolving all the cases of conscience, which their judgments seem to be unsatisfied in. I will tell you what are the impediments in your way, that are harder to be removed than all these difficulties, and yet must be overcome before you can bring men to be like true christians, rich in good works. 1. Most men are so sensual and selfish, that their own flesh is an insatiable gulf that devoureth all, and they have little or nothing to spare from it to good uses. It is better cheaply maintaining a family of temperate, sober persons, than one fleshly person that hath a whole litter of vices and lusts to be maintained: so much a year seemeth necessary to maintain their pride in needless curiosity and bravery, and so much a year to maintain their sensual sports and pleasures; and so much to please their throats or appetites, and to lay in provision for fevers, and dropsies, and coughs, and consumptions, and a hundred such diseases, which are the natural progeny of gluttony, drunkenness, and excess; and so much a year to maintain their idleness, and so of many other vices. But if one of these persons have the pride, and idleness, and gluttony, and sportfulness of wife, and children, and family also to maintain, as
- 61. well as their own, many thousand pounds a year perhaps may be too little. Many a conquering army hath been maintained at as cheap a rate, as such an army of lusts (or garrison at least) as keep possession of some such families, when all their luxury goeth for the honour of their family, and they glory in wearing the livery of the devil, the world, and the flesh (which they once renounced, and pretended to glory in nothing but the cross of Christ); and when they take care in the education of their children, that this entailed honour be not cut off from their families: no wonder if God's part be little from these men, when the flesh must have so much, and when God must stand to the courtesy of his enemies, and have but their leavings. I hope the nobility and gentry of England that are innocent herein, will not be offended with me, if I tell them that are guilty, that when I foresee their accounts, I think them to be the miserablest persons upon earth, that rob God, and rob the king of that which should defray the charges of government, and rob the church, and rob the poor, and rob their souls of all the benefits of good works, and all to please the devouring flesh. It is a dreadful thing to foresee with what horror they will give up their reckoning, when instead of so much in feeding and clothing the poor, and promoting the gospel, and the saving of men's souls, there will be found upon their account, so much in vain curiosities and pride, and so much in costly sports and pleasures, and so much in flesh- pleasing luxury and excess. The trick that they have got of late, to free themselves from the fears of this account, by believing that there will be no such day, will prove a short and lamentable remedy: and when that day shall come upon them unawares, their unbelief and pleasures will die together, and deliver them up to never-dying horror and despair. I have heard it often mentioned as the dishonour of France, that the third part of the revenues of so rich a kingdom should be devoted and paid to the maintaining of superstition: but if there be not many (and most) kingdoms in the world, where one half of their wealth is devoted to the flesh, and so to the devil, I should be glad to find myself herein mistaken: and judge you which is more disgraceful, to have half your estates given in sensuality to the devil, or a third part too ignorantly devoted to God! If men laid
- 62. out no more than needs upon the flesh, they might have the more for the service of God and of their souls. You cannot live under so much a year, as would maintain twice as many frugal, temperate, industrious persons, because your flesh must needs be pleased, and you are strangers to christian mortification and self-denial. Laertius tells us that Crates Thebanus put all the money into the banker's or usurer's hands, with this direction, That if his sons proved idiots it should all be paid to them, but if they proved philosophers it should be given to the poor; because philosophers can live upon a little, and therefore need little. So if we could make men mortified christians, they would need so little for themselves, that they would have the more to give to others, and to do good with. 2. Men do not seriously believe God's promises; that he will recompense them in this life (with better things) an hundred-fold, and in the world to come with life eternal! Matt. xix. 29. And that by receiving a prophet, or righteous man, they may have a prophet's or righteous man's reward, Matt. x. 41. And that a cup of cold water (when you have no better) given to one of Christ's little ones in the name of a disciple, shall not be unrewarded, Matt. x. 42. They believe not that heaven will pay for all, and that there is a life to come in which God will see that they be no losers. They think there is nothing certain but what they have in hand, and therefore they lay up a treasure upon earth, and rather trust to their estates than God; whereas if they verily believed that there is another life, and that judgment will pass on them on the terms described, Matt. xxv. they would more industriously lay up a treasure in heaven, Matt. vi. 20, and make themselves friends of the mammon of unrighteousness, and study how to be rich in good works, and send their wealth to heaven before them, and lay up a good foundation against the time to come, that they may lay hold upon eternal life, and then they would be ready to distribute, and willing to communicate, 1 Tim. vi. 17-19; Luke xvi. 9. They would then know how much they are beholden to God, that will not only honour them to be his stewards, but reward them for distributing his maintenance to his children, as if they had given so much of their own; they
- 63. would then see that it is they that are the receivers, and that giving is the surest way to be rich, when for transitory things (sincerely given) they may receive the everlasting riches. Then they would see that he that saveth his riches loseth them, and he that loseth them for Christ doth save them, and lay them up in heaven; and that it is more blessed to give than to receive; and that we should ourselves be laborious that we may have wherewith to support the weak, and to give unto the needy. Read Acts xx. 35; Eph. iv. 28; Prov. xxxi. 20, c. Then they would not be weary of well-doing, if they believed that, in due season, they shall reap if they faint not; but as they have opportunity, would do good to all men; but especially to them that are of the household of faith, Gal. vi. 9, 10. They would not forget to do good, and communicate, as knowing that with such sacrifices God is well pleased, Heb. xiii. 16. A true belief of the reward, would make men strive who should do most. 3. Another great hinderance is the want of love to God and our neighbours, to Christ and his disciples. If men loved Christ, they would not deal so niggardly with his disciples, when he has told them that he taketh all that they do to the least of them, (whom he calleth his brethren,) as done to himself, Matt. xxv.; x. 39, 40. If men loved their neighbours as themselves, I leave you to judge in what proportion and manner they would relieve them! Whether they would find money to lay out on dice and cards, and gluttonous feastings, on plays, and games, and pomp, and pride, while so many round about them are in pinching want. The destruction of charity or christian love is the cause that works of charity are destroyed. Who can look that the seed of the serpent, that hath an enmity against the holy seed, should liberally relieve them? or that the fleshly mind, which is enmity against God, should be ready to do good to the spiritual and holy servants of God? Gen. xv.; Rom. viii. 6-8; or that a selfish man should much care for any body but himself and his own? When love is turned into the hatred of each other, upon the account of our partial interests and opinions; and when we are like men in war, that think he is the bravest, most
- 64. deserving man that hath killed most; when men have bitter, hateful thoughts of one another, and set themselves to make each other odious, and to ruin them, that they may stand the faster, and think that destroying them is good service to God; who can look for the fruits of love from damnable uncharitableness and hatred; or that the devil's tree should bring forth holy fruit to God? 4. And then (when love is well spoken of by all, even its deadly enemies) lest men should see their wickedness and misery, (and is it not admirable that they see it not?) the devil hath taught them to play the hypocrite, and make themselves a religion which costs them nothing, without true christian love and good works, that they may have something to quiet and cheat their consciences with. One man drops now and then an inconsiderable gift, and another oppresseth, and hateth, and destroyeth (and slandereth and censureth, that he may not be thought to hate and ruin without cause); and when they have done, they wipe their mouths with a few hypocritical prayers or good words, and think they are good christians, and God will not be avenged on them. One thinks that God will save him because he is of this church, and another because he is of another church. One thinks to be saved because he is of this opinion and party in religion, and another because he is of that. One thinks he is religious because he saith his prayers this way, and another because he prayeth another way. And thus dead hypocrites, whose hearts were never quickened with the powerful love of God, to love his servants, their neighbours, and enemies, do persuade themselves that God will save them for mocking and flattering him with the service of their deceitful lips; while they want the love of God, which is the root of all good, and are possessed with the love of money, which is the root of all evil, 1 Tim. vi. 10, and are lovers of pleasures more than of God, 2 Tim. iii. 4. They will join themselves forwardly to the cheap and outside actions of religion; but when they hear much less than One thing thou yet wantest: sell all that thou hast and distribute to the poor, and thou shalt have a treasure in heaven:—they are very sorrowful, because they are very rich, Luke xviii. 22, 23. Such a fruitless love
- 65. as they had to others, James ii. such a fruitless religion they have as to themselves. For pure religion and undefiled before God, is to visit the fatherless and widows in their adversity, and to keep yourselves unspotted from the world, James i. 27. See 1 John ii. 15; iii. 17, Whoso hath this world's goods, and seeth his brother have need, and shutteth up his bowels of compassion from him, how dwelleth the love of God in him? There are three texts that describe the case of sensual, uncharitable gentlemen. 1. Luke xvi. A rich man clothed in purple and silk, (for so, as Dr. Hammond noteth, it should rather be translated,) and fared sumptuously every day, you know the end of him. 2. Ezek. xvi. 49. Sodom's sin was pride, fulness of bread, and abundance of idleness, neither did she strengthen the hand of the poor and needy. 3. James v. 1-7. Go to now, ye rich men, weep and howl for the miseries that shall come upon you.—Ye have lived in pleasure on earth, and been wanton; ye have nourished your hearts, as in (or for) the day of slaughter.—Ye have condemned and killed the just, and he doth not resist you—. And remember Prov. xxi. 13, Whoso stoppeth his ears at the cry of the poor, he also shall cry himself and shall not be heard. And James ii. 13, He shall have judgment without mercy that showed no mercy, and mercy rejoiceth against judgment. Yea, in this life it is oft observable that, Prov. xi. 24, There is that scattereth, and yet increaseth, and there is that withholdeth more than is meet, but it tendeth to poverty. Tit. 2. Directions for Works of Charity. Direct. I. Love God, and be renewed to his image; and then it will be natural to you to do good; and his love will be in you a fountain of good works. Direct. II. Love your neighbours, and it will be easy to you to do them all the good you can; as it is to do good to yourselves, or children, or dearest friends.
- 66. Direct. III. Learn self-denial, that selfishness may not cause you to be all for yourselves, and be Satan's law of nature in you, forbidding you to do good to others. Direct. IV. Mortify the flesh, and the vices of sensuality: pride and curiosity, gluttony and drunkenness, are insatiable gulfs, and will devour all, and leave but little for the poor: though there be never so many poor families which want bread and clothing, the proud person must first have the other silk gown, or the other ornaments which may set them out with the forwardest in the mode and fashion; and this house must first be handsomer built, and these rooms must first be neatlier furnished; and these children must first have finer clothes: let Lazarus lie never so miserable at the door, the sensualist must be clothed in purple and silk, and fare deliciously and sumptuously daily, Luke xvi. The glutton must have the dish and cup which pleaseth his appetite, and must keep a full table for the entertainment of his companions that have no need. These insatiable vices are like swine and dogs, that devour all the children's bread. Even vain recreations and gaming shall have more bestowed on them, than church or poor (as to any voluntary gift). Kill your greedy vices once, and then a little will serve your turns, and you may have wherewith to relieve the needy, and do that which will be better to you at your reckoning day. Direct. V. Let not selfishness make your children the inordinate objects of your charity and provision, to take up that which should be otherwise employed. Carnal and worldly persons would perpetuate their vice, and when they can live no longer themselves, they seem to be half alive in their posterity, and what they can no longer keep themselves, they think is best laid up for their children to feed them as full, and make them as sensual and unhappy as themselves. So that just and moderate provisions will not satisfy them; but their children's portions must be as much as they can get, and almost all their estates are sibi et suis, for themselves and theirs: and this pernicious vice is as destructive to good works, as almost any in the world. That God who hath said that he is worse than an infidel who provideth not for his own family, will judge many
- 67. thousands to be worse than christians, and than any that will be saved must be, who make their families the devourers of all which should be expended upon other works of charity. Direct. VI. Take it as the chiefest extrinsical part of your religion to do good; and make it the trade or business of your lives, and not as a matter to be done on the by. James i. 27, Pure religion and undefiled before God and the Father is this, to visit the fatherless and widows in their affliction, and to keep himself unspotted from the world. If we are created for good works, Eph. ii. 10; and redeemed and purified to be zealous of good works, Tit. ii. 14; and must be judged according to such works, Matt. xxv.; then certainly it should be our chiefest daily care and diligence, to do them with all our hearts and abilities. And as we keep a daily account of our own and our servants' business in our particular callings, so should we much more of our employment of our Master's talents in his service; and if a heathen prince could say with lamentation, Alas, I have lost a day! if a day had passed in which he had done no one good, how much more should a christian, who is better instructed to know the comforts and rewards of doing good! Direct. VII. Give not only out of your superfluities, when the flesh is glutted with as much as it desireth; but labour hard in your callings, and be thrifty and saving from all unnecessary expenses, and deny the desires of ease and fulness, and pride and curiosity, that you may have the more to do good with. Thriftiness for works of charity is a great and necessary duty, though covetous thriftiness for the love of riches be a great sin. He that wasteth one half of his master's goods through slothfulness or excesses, and then is charitable with the other half, will make but a bad account of his stewardship. Much more he that glutteth his own and his family's and retainers' fleshly desires first, and then giveth to the poor only the leavings of luxury, and so much as their fleshly lusts can spare. It is a dearer, a laborious and a thrifty charity, that God doth expect of faithful stewards.
- 68. Direct. VIII. Delay not any good work which you have present ability and opportunity to perform. Delay signifieth unwillingness or negligence. Love and zeal are active and expeditious; and delay doth frequently frustrate good intentions. The persons may die that you intend to do good to; or you may die, or your ability and opportunities may cease; that may be done to-day which cannot be done to-morrow. The devil is not ignorant of your good intentions, and he will do all that possibly he can to make them of no effect; and the more time you give him, the more you enable him to hinder you. You little foresee what abundance of impediments he may cast before you; and so make that impossible which once you might have done with ease. Prov. iii. 28, Say not to thy neighbour, Go and come again, and to-morrow I will give, when thou hast it by thee. Prov. xxvii. 1, Boast not thyself of to-morrow, for thou knowest not what a day may bring forth. Direct. IX. Distrust not God's providence for thy own provision. An unbelieving man will needs be a God to himself, and trust himself only for his provisions, because indeed he cannot trust God. But you will find that your labour and care are vain, or worse than vain, without God's blessing. Say not distrustfully, What shall I have myself when I am old? Though I am not persuading you to make no provision, or to give away all; yet I must tell you, that it is exceeding folly to put off any present duty, upon distrust of God, or expectation of living to be old. He that over-night said, I have enough laid up for many years, did quickly hear, Thou fool, this night shall thy soul be required of thee; and whose then shall the things be which thou hast provided? Luke xii. 20. Rather obey that, Eccles. ix. 10, Whatsoever thy hand findeth to do, do it with thy might: for there is no work, nor device, nor knowledge, nor wisdom in the grave whither thou goest. Do you think there is not a hundred thousand whose estates are now consumed in the flames of London, who could wish that all that had been given to pious or charitable uses? Do but believe from the bottom of your hearts, that he that hath pity on the poor, lendeth to the Lord, and that which he layeth out he will pay him again, Prov. xix. 17. And that, Matt. x. 40-42, He
- 69. that receiveth you, receiveth me, and he that receiveth me, receiveth him that sent me: he that receiveth a prophet in the name of a prophet, shall receive a prophet's reward; and he that receiveth a righteous man, in the name of a righteous man, shall receive a righteous man's reward: and whosoever shall give to drink to one of these little ones, a cup of cold water only (i. e. when he hath no better) in the name of a disciple, verily I say unto you, he shall in no wise lose his reward. I say, believe this, and you will make haste to give while you may, lest your opportunity should overslip you. Direct. X. What you cannot do yourselves, provoke others to do who are more able: Provoke one another to love and to good works. Modesty doth not so much forbid you to beg for others as for yourselves. Some want but information to draw them to good works: and some that are unwilling, may be urged to it, to avoid the shame of uncharitableness: and though such giving do little good to themselves, it may do good to others. Thus you may have the reward when the cost is another's as long as the charity is yours. Direct. XI. Hearken to no doctrine which is an enemy to charity or good works; nor yet which teacheth you to trust in them for more than their proper part. He that ascribeth to any of his own works, that which is proper to Christ, doth turn them into heinous sin. And he that ascribeth not to them all that which Christ ascribeth to them, is a sinner also. And whatever ignorant men may prate, the time is coming, when neither Christ without our charity, nor our charity without Christ, (but in subordination to him,) will either comfort or save our souls.
- 70. [173] See the Preface to my book, called, The Crucifying of the World.
- 71. CHAPTER XXXI. CASES AND DIRECTIONS ABOUT CONFESSING SINS AND INJURIES TO OTHERS. Tit. 1. Cases about Confessing Sins and Injuries to others. Quest. I. In what cases is it a duty to confess wrongs to those that we have wronged? Answ. 1. When in real injuries you are unable to make any restitution, and therefore must desire forgiveness, you cannot well do it without confession. 2. When you have wronged a man by a lie, or by false witness, or that he cannot be righted till you confess the truth. 3. When you have wronged a man in his honour or fame, where the natural remedy is to speak the contrary, and confess the wrong. 4. When it is necessary to cure the revengeful inclination of him whom you have wronged, or to keep up his charity, and so to enable him to love you, and forgive you. 5. Therefore all known wrongs to another must be confessed, except when impossibility, or some ill effect which is greater than the good, be like to follow. Because all men are apt to abate their love to those that injure them, and therefore all have need of this remedy. And we must do our part to be forgiven by all whom we have wronged. Quest. II. What causes will excuse us from confessing wrongs to others? Answ. 1. When full recompence may be made without it, and no forgiveness of the wrong is necessary from the injured, nor any of the aforesaid causes require it. 2. When the wrong is secret and not known to the injured party, and the confessing of it would but trouble his mind, and do him more harm than good. 3. When the injured party is so implacable and inhumane that he would make use
- 72. of the confession to the ruin of the penitent, or to bring upon him greater penalty than he deserveth. 4. When it would injure a third person who is interested in the business, or bring them under oppression and undeserved misery. 5. When it tendeth to the dishonour of religion, and to make it scorned because of the fault of the penitent confessor. 6. When it tendeth to set people together by the ears, and breed dissension, or otherwise injure the commonwealth or government. 7. In general, it is no duty to confess our sin to him that we have wronged, when, all things considered, it is like in the judgment of the truly wise, to do more hurt than good: for it is appointed as a means to good, and not to do evil. Quest. III. If I have had a secret thought or purpose to wrong another, am I bound to confess it, when it was never executed? Answ. 1. You are not bound to confess it to the party whom you intended to wrong, as any act of justice to make him reparation; nor to procure his forgiveness to yourself: because it was no wrong to him indeed, nor do thoughts and things secret come under his judgment, and therefore need not his pardon. 2. But it is a sin against God, and to him you must confess it. 3. And by accident, finis gratia, you must confess it to men, in case it be necessary to be a warning to others, or to the increase of their hatred of sin, or their watchfulness, or to exercise your own humiliation, or prevent a relapse, or to quiet your conscience, or in a word, when it is like to do more good than hurt. Quest. IV. To whom, and in what cases, must I confess to men my sins against God, and when not? Answ. The cases about that confession which belongeth to church discipline, belongeth to the second part; and therefore shall here be passed by. But briefly and in general, I may answer the question thus: There are conveniences and inconveniences to be compared together, and you must make your choice accordingly. The reasons which may move you to confess your sins to another are these: 1. When another hath sinned with you, or persuaded or drawn you to it, and must be brought to repentance with you. 2. When your
- 73. conscience hath in vain tried all other fit means for peace or comfort, and cannot obtain it, and there is any probability of such advice from others as may procure it. 3. When you have need of advice to resolve your conscience, whether it be sin or not, or of what degree, or what you are obliged to in order to forgiveness. 4. When you have need of counsel to prevent the sin for the time to come, and mortify the habit of it. The inconveniences which may attend it, are such as these: 1. You are not certain of another's secrecy; his mind may change, or his understanding fail, or he may fall out with you, or some great necessity may befall him to drive him to open what you told him. 2. Then whether your shame or loss will not make you repent it, should be foreseen. 3. And how far others may suffer in it. 4. And how far it will reflect dishonour on religion. All things being considered on both sides, the preponderating reasons must prevail. Tit. 2. Directions about Confessing Sin to others. Direct. I. Do nothing which you are not willing to confess, or which may trouble you much, if your confession should be opened. Prevention is the easiest way: and foresight of the consequents should make a wise man still take heed. Direct. II. When you have sinned or wronged any, weigh well the consequents on both sides before you make your confession: that you may neither do that which you may wish undone again, nor causelessly refuse your duty: and that inconveniences foreseen may be the better undergone when they cannot be avoided. Direct. III. When a well-informed conscience telleth you that confession is your duty, let not self-respects detain you from it, but do it whatever it may cost you. Be true to conscience, and do not wilfully put off your duty. To live in the neglect of a known duty, is to live in a known sin: which will give you cause to question your sincerity, and cause more terrible effects in your souls, than the inconveniences of confession could ever have been.
- 74. Direct. IV. Look to your repentance that it be deep and absolute, and free from hypocritical exceptions and reserves. For half and hollow repentance will not carry you through hard and costly duties. But that which is sincere, will break over all: it will make you so angry with yourselves and sins, that you will be as inclined to take shame to yourselves in an honest revenge, as an angry man is to bring shame upon his adversary. We are seldom over-tender of a man's reputation whom we fall out with: and repentance is a falling out with ourselves. We can bear sharp remedies, when we feel the pain, and perceive the mortal danger of the disease: and repentance is such a perception of our pain and danger. We will not tenderly hide a mortal enemy, but bring him to the most open shame: and repentance causeth us to hate sin as our mortal enemy. It is want of repentance that maketh men so unwilling to make a just confession. Direct. V. Take heed of pride, which maketh men so tender of their reputation, that they will venture their souls to save their honour: men call it bashfulness, and say they cannot confess for shame; but it is pride that maketh them so much ashamed to be known by men to be offenders, while they less fear the eye and judgment of the Almighty. Impudence is a mark of a profligate sinner; but he that pretendeth shame against his duty, is foolishly proud; and should be more ashamed to neglect his duty, and continue impenitent in his sin. A humble person can perform a self-abasing, humbling duty. Direct. VI. Know the true uses of confession of sin, and use it accordingly. Do it with a hatred of sin, to express yourselves implacable enemies to it: do it to repair the wrong which you have done to others, and the dishonour you have done to the christian religion, and to warn the hearers to take heed of sin and temptation by your fall; it is worth all your shame, if you save one sinner by it from his sin: do it to lay the greater obligation upon yourselves for the future, to avoid the sin and live more carefully; for it is a double shame to sin after such humbling confessions.
- 75. CHAPTER XXXII. CASES AND DIRECTIONS ABOUT SATISFACTION AND RESTITUTION. Tit. 1. Cases of Conscience about Satisfaction and Restitution. Quest. I. When is it that proper restitution must be made, and when satisfaction? and what is it? Answ. Restitution properly is ejusdem, of the same thing, which was detained or taken away. Satisfaction is solutio æquivalentis, vel tantidem, alias indebiti, that which is for compensation or reparation of loss, damage, or injury; being something of equal value or use to the receiver. Primarily res ipsa debetur, restitution is first due, where it is possible; but when that is unavoidably hindered or forbidden by some effectual restraint, satisfaction is due. Whilst restitution of the same may be made, we cannot put off the creditor or owner with that which is equivalent without his own consent; but by his consent we may at any time. And to the question, What is due satisfaction? I answer, that when restitution may be made, and he that should restore doth rather desire the owner to accept some other thing in compensation, there that proportion is due satisfaction which both parties agree upon. For if it be above the value it was yet voluntarily given, and the payer might have chosen: and if it be under the value, it was yet voluntarily accepted, and the receiver might have chosen. But if restitution cannot be made, or not without some greater hurt to the payer than the value of the thing, there due satisfaction is that which is of equal value and use to the receiver; and if he will not be satisfied with it, he is unjust, and it is quoad valorem rei et debitum solventis, full satisfaction, and he is not (unless by some other accident) bound to give any more; because it is not another's unrighteous will that he is obliged to fulfil, but a
- 76. Why did they restore fourfold? debt which is to be discharged. But here you must distinguish betwixt satisfaction in commutative justice, for a debt or injury, and satisfaction in distributive, governing justice, for a fault or crime. The measure of the former satisfaction, is so much as may compensate the owner's loss; not only so much as the thing was worth to another, but what it was worth to him: but the measure of the latter satisfaction, is so much as may serve the ends of government instead of actual obedience; or so much as will suffice to the ends of government, to repair the hurts which the crime hath done, or avoid what it would do. And here you may see the answer to that question, Why a thief was commanded to restore fourfold, by the law of Moses; for in that restitution there was a conjunction of both these sorts of satisfaction, both in point of commutative and distributive justice: so much as repaired the owner's loss was satisfaction to the owner for the injury: the rest was all satisfaction to God and the commonwealth for the public injury that came by the crime or violation of the law. Other answers are given by some, but this is the plain and certain truth. Quest. II. How far is restitution or satisfaction necessary? Answ. As far as acts of obedience to God and justice to man are necessary: that is, 1. As a man that repenteth truly of sin against God, may be saved without external obedience, if you suppose him cut off by death immediately upon his repenting, before he hath any opportunity to obey; so that the animus obediendi is absolutely necessary, and the actus obediendi if there be opportunity: so is it here, the animus restituendi, or true resolution or willingness to restore, is ever necessary to the sincerity of justice and repentance in the person, as well as necessary necessitate præcepti; and the act of restitution primarily, and of satisfaction secondarily, is necessary, if; there be time and power: I say necessary always as a duty, necessitate præcepti; and necessary necessitate medii, as a condition of pardon and salvation, so far as they are necessary acts
- 77. of true repentance and obedience, as other duties are: that is, as a true penitent may in a temptation omit prayer or church communion, but yet hath always such an habitual inclination to it, as will bring him to it when he hath opportunity by deliberation to come to himself; and as in the same manner a true penitent may omit a work of charity or mercy, but not give over such works; even so is it in this case of restitution and satisfaction. Quest. III. Who are they that are bound to make restitution or satisfaction? Answ. 1. Every one that possesseth and retaineth that which is indeed another man's, and hath acquired no just title to it himself, must make restitution. Yet so, that if he came lawfully by it (as by finding, buying, or the like) he is answerable for it only upon the terms in those titles before expressed. But if he came unlawfully by it, he must restore it with all damages. The cases of borrowers and finders are before resolved. He that keepeth a borrowed thing longer than his day, must return it with the damage. He that loseth a thing which he borrowed, must make satisfaction, unless in cases where the contract, or common usage, or the quality of the thing, excuseth him. 2. He that either by force, or fraud, or negligence, or any injustice, doth wrong to another, is bound to make him a just compensation, according to the proportion of the guilt and the loss compared together; for neither of them is to be considered alone. If a servant neglect his master's business, and it fall out that no loss followeth it, he is bound to confess his fault, but not to pay for a loss which might have been, but was not. And if a servant by some such small and ordinary negligence, which the best servants are guilty of, should bring an exceeding great damage upon his master, (as by dropping asleep to burn his house, or by an hour's delay which seemed not very dangerous, to frustrate some great business) he is obliged to reparation as well as to confession; but not to make good all that is lost, but according to the proportion of his fault. But he that by oppression or robbery taketh that which is another's, or bringeth any damage to him, or by slander, false-witness, or any such unrighteous means, is bound to make a fuller satisfaction; and
- 78. those that concur in the injury, being accessories, are bound to satisfy. As, 1. Those that teach or command another to do it. 2. Those who send a commission, or authorize another to do it. 3. Those who counsel, exhort, or persuade another to do it. 4. Those who by consenting are the causes of it. 5. Those who co-operate and assist in the injury knowingly and voluntarily. 6. Those who hinder it not when they could and were obliged to do it. 7. Those who make the act their own, by owning it, or consenting afterward. 8. Those who will not reveal it afterward, that the injured party may recover his own, when they are obliged to reveal it. But a secret consent which no way furthered the injury, obligeth none to restitution, but only to repentance; because it did no wrong to another, but it was a sin against God. Quest. IV. To whom must restitution or satisfaction be made? Answ. 1. To the true owner, if he be living and to be found, and it can be done. 2. If that cannot be, then to his heirs, who are the possessors of that which was his. 3. If that cannot be, then to God himself, that is, to the poor, or unto pious uses; for the possessor is no true owner of it; and therefore where no other owner is found, he must discharge himself so of it, to the use of the highest and principal Owner, as may be most agreeable to his will and interest. [174] Quest. V. What restitution should he make who hath dishonoured his governors or parents? Answ. He is bound to do all that he can to repair their honour, by suitable means; and to confess his fault, and crave their pardon. Quest. VI. How must satisfaction be made for slanders, lies, and defaming of others? Answ. By confessing the sin, and unsaying what was said, not only as openly as it was spoken, but as far as it is since carried on by others, and as far as the reparation of your neighbour's good name requireth, if you are able.
- 79. Quest. VII. What reparation must they make who have tempted others to sin, and hurt their souls? Answ. 1. They must do all that is in their power to recover them from sin, and to do good to their souls. They must go to them, and confess and lament the sin, and tell them the evil and danger of it, and incessantly strive to bring them to repentance. 2. They must make reparation to the Lord of souls, by doing all the good they can to others, that they may help more than they have hurt. Quest. VIII. What reparation can or must be made for murder or manslaughter? Answ. By murder there is a manifold damage inferred: 1. God is deprived of the life of his servant. 2. The person is deprived of his life. 3. The king is deprived of a subject. 4. The commonwealth is deprived of a member. 5. The friends and kindred of the dead are deprived of a friend. 6. And perhaps also damnified in their estates. All these damages cannot be fully repaired by the offender; but all must be done that can be done. 1. Of God he can only beg pardon, upon the account of the satisfactory sacrifice of Christ; expressing true repentance as followeth. 2. To the person murdered no reparation can be made. 3. To the king and commonwealth, he must patiently yield up his life, if they sentence him to death, and without repining, and think it not too dear to become a warning to others, that they sin not as he did. 4. To disconsolate friends no reparation can be made; but pardon must be asked. 5. The damage of heirs, kindred, and creditor, must be repaired by the offender's estate, as far as he is able. Quest. IX. Is a murderer bound to offer himself to death, before he is apprehended? Answ. Yes, in some cases: as, 1. When it is necessary to save another who is falsely accused of the crime. 2. Or when the interest of the commonwealth requireth it. But otherwise not; because an offender may lawfully accept of mercy, and nature teacheth him to desire his own preservation: but if the question be, When doth the
- 80. interest of the commonwealth require it? I think much oftener than it is done: as the common interest requireth that murderers be put to death, when apprehended; so it requireth that they may not frequently and easily be hid, or escape by secrecy or flight; for then it would imbolden others to murder: whereas when few escape, it will more effectually deter men. If therefore any murderer's conscience shall constrain him in true repentance, voluntarily to come forth and confess his sin, and yield up himself to justice, and exhort others to take heed of sinning as he did, I cannot say that he did any more than his duty in so doing; and indeed I think that it is ordinarily a duty, and that ordinarily the interest of the commonwealth requireth it; though in some cases it may be otherwise. The execution of the laws against murder, is so necessary to preserve men's lives, that I do not think that self-preservation alone will allow men to defeat the commonwealth of so necessary a means of preserving the lives of many, to save the life of one, who hath no right to his own life, as having forfeited it. If to shift away other murderers from the hand of justice be a sin, I cannot see but that it is so ordinarily to do it for oneself: only I think that if a true penitent person have just cause to think that he may do the commonwealth more service by his life than by his death, that then he may conceal his crime or fly; but otherwise not. Quest. X. Is a murderer bound to do execution on himself, if the magistrate upon his confession do not? Answ. No: because it is the magistrate who is the appointed judge of the public interest, and what is necessary to its reparation, and hath power in certain cases to pardon: and though a murderer may not ordinarily strive to defeat God's laws and the commonwealth, yet he may accept of mercy when it is offered him. Quest. XI. What satisfaction is to be made by a fornicator or adulterer? Answ. Chastity cannot be restored, nor corrupted honour repaired. But, 1. If it was a sin by mutual consent, the party that you sinned with must by all importunity be solicited to repentance; and the sin
- 81. must be confessed, and pardon craved for tempting them to sin. 2. Where it can be done without a greater evil than the benefit will amount to, the fornicators ought to join in marriage, Exod. xxii. 16. 3. Where that cannot be, the man is to put the woman into as good a case for outward livelihood, as she would have been in if she had not been corrupted by him; by allowing her a proportionable dowry, Exod. xxii. 17; and the parents' injury to be recompensed, Deut. xxii. 28, 29. 4. The child's maintenance also is to be provided for by the fornicator. That is, 1. If the man by fraud or solicitation induced the woman to the sin, he is obliged to all as aforesaid. 2. If they sinned by mutual forwardness and consent, then they must jointly bear the burden; yet so that the man must bear the greater part, because he is supposed to be the stronger and wiser to have resisted the temptation. 3. If the woman importuned the man, she must bear the more: but yet he is responsible to parents and others for their damages, and in part to the woman herself, because he was the stronger vessel, and should have been more constant: and volenti non fit injuria, is a rule that hath some exceptions. Quest. XII. In what case is a man excused from restitution and satisfaction? Answ. 1. He that is utterly disabled cannot restore or satisfy. 2. He that is equally damnified by the person to whom he should restore, is excused in point of real equity and conscience, so be it that the reasons of external order and policy oblige him not. For though it may be his sin (of which he is to repent) that he hath equally injured the other, yet it requireth confession, rather than restitution or satisfaction, unless he may also expect satisfaction from the other. Therefore if you owe a man an hundred pounds, and he owe you as much and will not pay you, you are not bound to pay him, unless for external order sake, and the law of the land. 3. If the debt or injury be forgiven, the person is discharged. 4. If nature or common custom do warrant a man to believe that no restitution or satisfaction is expected, or that the injury is forgiven, though it be not mentioned, it will excuse him from restitution or satisfaction: as if children or friends have taken some trifle, which they may
- 82. presume the kindness of a parent or friend will pass over, though it be not justifiable. Quest. XIII. What if the restitution will cost the restorer far more than the thing is worth? Answ. He is obliged to make satisfaction, instead of restitution. Quest. XIV. What if the confessing of the fault may engage him that I must restore to, so that he will turn it to my infamy or ruin? Answ. You may then conceal the person, and send him satisfaction by another hand; or you may also conceal the wrong itself, and cause satisfaction to be made him, as by gift, or other way of payment. Tit. 2. Directions about Restitution and Satisfaction. Direct. I. Foresee the trouble of restitution, and prevent it. Take heed of covetousness, which would draw you into such a snare. What a perplexed case are some men in, who have injured others so far as that all they have will scarce make them due satisfaction! Especially public oppressors, who injure whole nations, countries, or communities: and unjust judges, who have done more wrong perhaps in one day or week than all their estates are worth: and unjust lawyers, who plead against a righteous cause: and false witnesses, who contribute to the wrong: and unjust juries, or any such like: also oppressing landlords; and soldiers that take men's goods by violence; and deceitful tradesmen, who live by injuries. In how sad a case are all these men! Direct. II. Do nothing which is doubtful, if you can avoid it, lest it should put you upon the trouble of restitution. As in case of any doubtful way of usury or other gain, consider, that if it should hereafter appear to you to be unlawful, and so you be obliged to restitution, (though you thought it lawful at the taking of it,) what a snare then would you be in, when all that use must be repaid! And so in other cases.
- 83. Direct. III. When really you are bound to restitution or satisfaction, stick not at the cost or suffering, be it never so great, but be sure to deal faithfully with God and conscience. Else you will keep a thorn in your hearts, which will smart and fester till it be out: and the ease of your consciences will bear the charge of your costliest restitution. Direct. IV. If you be not able in your lifetime to make restitution, leave it in your wills as a debt upon your estates; but never take it for your own. Direct. V. If you are otherwise unable to satisfy, offer your labour as a servant to him to whom you are indebted; if at least by your service you can make him a compensation. Direct. VI. If you are that way unable also, beg of your friends to help you, that charity may enable you to pay the debt. Direct. VII. But if you have no means at all of satisfying, confess the injury and crave forgiveness, and cast yourself on the mercy of him whom you have injured. [174] Heb. v. 23; 1 Sam xii. 3; Neh. v. 11; Numb. v. 8; Luke xix. 8.
- 84. CHAPTER XXXIII. CASES AND DIRECTIONS ABOUT OUR OBTAINING PARDON FROM GOD. Tit. 1. Cases of Conscience about Obtaining Pardon of Sin from God. Quest. I. Is there pardon to be had for all sin without exception, or not? Answ. 1. There is no pardon procured or offered, for the final non- performance of the conditions of pardon; that is, for final impenitency, unbelief, and ungodliness. 2. There is no pardon for any sin, without the conditions of pardon, that is, without true faith and repentance, which is our conversion from sin to God. 3. And if there be any sin which certainly excludeth true repentance to the last, it excludeth pardon also; which is commonly taken to be the case of blasphemy against the Holy Ghost; of which I have written at large in my Treatise against Infidelity. But, 1. All sin, except the final non-performance of the conditions of pardon, is already conditionally pardoned in the gospel; that is, if the sinner will repent and believe. No sin is excepted from pardon to penitent believers. 2. And all sin is actually pardoned to a true penitent believer. Quest. II. What if a man do frequently commit the same heinous sin; may he be pardoned? Answ. Whilst he frequently committeth it (being a mortal sin) he doth not truly repent of it; and whilst he is impenitent he is unpardoned: but if he be truly penitent, his heart being habitually and actually turned from the sin, it will be forgiven him; but not till he thus forsake it.
- 85. Welcome to our website – the perfect destination for book lovers and knowledge seekers. We believe that every book holds a new world, offering opportunities for learning, discovery, and personal growth. That’s why we are dedicated to bringing you a diverse collection of books, ranging from classic literature and specialized publications to self-development guides and children's books. More than just a book-buying platform, we strive to be a bridge connecting you with timeless cultural and intellectual values. With an elegant, user-friendly interface and a smart search system, you can quickly find the books that best suit your interests. Additionally, our special promotions and home delivery services help you save time and fully enjoy the joy of reading. Join us on a journey of knowledge exploration, passion nurturing, and personal growth every day! ebookbell.com