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Markov Processes Semigroups and Generators 1st
Edition Vassili N. Kolokoltsov Digital Instant Download
Author(s): Vassili N. Kolokoltsov
ISBN(s): 9783110250107, 3110250101
Edition: 1
File Details: PDF, 2.39 MB
Year: 2011
Language: english

De Gruyter Studies in Mathematics 38
Editors
Carsten Carstensen, Berlin, Germany
Nicola Fusco, Napoli, Italy
Fritz Gesztesy, Columbia, USA
Niels Jacob, Swansea, United Kingdom
Karl-Hermann Neeb, Erlangen, Germany

Vassili N. Kolokoltsov
Markov Processes,
Semigroups and Generators
De Gruyter

Mathematical Subject Classification 2010: Primary: 60-01, 60-02; Secondary: 60G51, 60G52,
60H10, 60H30, 60G50, 60J25, 60K35, 60F17, 81Q05, 81S40.
ISBN 978-3-11-025010-7
e-ISBN 978-3-11-025011-4
ISSN 0179-0986
Library of Congress Cataloging-in-Publication Data
Kolokol’tsov, V. N. (Vasilii Nikitich)
Markov processes, semigroups, and generators / by Vassili N.
Kolokoltsov.
p. cm. ⫺ (De Gruyter studies in mathematics ; 38)
Includes bibliographical references and index.
ISBN 978-3-11-025010-7 (alk. paper)
1. Markov processes. 2. Semigroups. 3. Group theory ⫺
Generators. I. Title.
QA274.7.K65 2011
519.2133⫺dc22
2010050783
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.
” 2011 Walter de Gruyter GmbH & Co. KG, Berlin/New York
Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de
Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen
⬁ Printed on acid-free paper
Printed in Germany
www.degruyter.com

To the memory of my parents,
Nikita and Svetlana Kolokoltsov,
who helped me much to become a scientist

Preface
Markov processes represent a universal model for a large variety of real life random
evolutions. The wide flow of new ideas, tools, methods and applications constantly
pours into the ever-growing stream of research on Markov processes that rapidly
spreads over new fields of natural and social sciences, creating new streamlined log-
ical paths to its turbulent boundary. Even if a given process is not Markov, it can be
often inserted into a larger Markov one (Markovianization procedure) by including
the key historic parameters into the state space.
Markov processes are described probabilistically by the distributions on their tra-
jectories (often specified by stochastic differential equations) and analytically by the
Markov semigroups that specify the evolution of averages and arise from the so-
lutions to a certain class of integro-differential (or pseudo-differential) equations,
which is distinguished by the preservation of the positivity property (probabilities
are positive). Thus the whole development stands on two legs: stochastic analysis
(with tools such as the martingale problem, stochastic differential equations, conver-
gence of measures on Skorohod spaces), and functional analysis (weighted Sobolev
spaces, pseudo-differential operators, operator semigroups, methods of Hilbert and
Fock spaces, Fourier analysis).
The aim of the monograph is to give a concise (but systematic and self-contained)
exposition of the essentials of Markov processes (highly nontrivial, but conceptually
excitingly rich and beautiful), together with recent achievements in their construc-
tions and analysis, stressing specially the interplay between probabilistic and analytic
tools. The main point is in the construction and analysis of Markov processes from the
‘physical picture’ – a formal pre-generator that specifies the corresponding evolution-
ary equation (here the analysis really meets probability) paying particular attention to
the universal models (analytically – general positivity-preserving evolutions), which
go above standard cases (e.g. diffusions and jump-type processes).
The introductory Part I is an enlarged version of the one-semester course on Brown-
ian motion and its applications given by the author to the final year mathematics and
statistics students of Warwick University. In this course, Browninan motion was stud-
ied not only as the simplest continuous random evolution, but as a basic continuous
component of complex processes with jumps. Part I contains mostly well-known ma-
terial, though written and organized with a point of view that anticipates further de-
velopments. In some places it provides more general formulations than usual (as with
the duality theorem in Section 1.9 or with the Holtzmark distributions in Section 1.5)
and new examples (as in Section 2.11).
Part II is based mainly on the author’s research. To facilitate the exposition, each
chapter of Part II is composed in such a way that it can be read almost independently

viii Preface
of others, and it ends with a section containing comments on bibliography and related
topics. The main results concern:
(i) various constructions and basic continuity properties of Markov processes, includ-
ing processes stopped or killed at the boundary (as well as related boundary points
classiﬁcation and sensitivity analysis),
(ii) in particular, heat kernel estimates for stable-like processes,
(iii) limiting processes for position-dependent continuous time random walks (ob-
tained by a random time change from the Markov processes) and related fractional (in
time) dynamics,
(iv) the rigorous Feynman path-integral representation for the solutions of the basic
equations of quantum mechanics, via jump-type Markov processes.
We also touch upon the theory of stochastic monotonicity, stochastic scattering,
stochastic quasi-classical (also called small diffusion) asymptotics, and stochastic
control. An important development of the methods discussed here is given by the
theory of nonlinear Markov processes (including processes on manifolds) presented
in the author’s monograph [196]. They are brieﬂy introduced at the end of Chapter 5.
It is worth pointing out the directions of research closely related to the main topic
of this book, but not touched here. These are Dirichlet forms, which can be used for
constructing Markov processes instead of generators, Mallivin calculus, which is a
powerful tool for proving various regularity properties for transition probabilities, log-
Sobolev inequalities, designed to systematically analyze the behavior of the processes
for large times, and processes on manifolds. There exists an extensive literature on
each of these subjects.
The book is meant to become a textbook and a monograph simultaneously, taking
more features of the latter as the exposition advances. I include some exercises, their
weight being much more sound at the beginning. The exercises are supplied with
detailed hints and are meant to be doable with the tools discussed in the book. The
exposition is reasonably self-contained, with pre-requisites being just the standard
math culture (basic analysis and linear algebra, metric spaces, Hilbert and Banach
spaces, Lebesgue integration, elementary probability). We shall start slowly from
the prerequisites in probability and stochastic processes, omitting proofs if they are
well presented in university text books and not very instructive for our purposes, but
stressing ideas and technique that are specially relevant. Streamlined logical paths
are followed to the main ideas and tools for the most important models, by-passing
wherever possible heavy technicalities (say, by working with Lévy processes instead
of general semi-martingales, or with left-continuous processes instead of predictable
ones).
A methodological aspect of the presentation consists in often showing various per-
spectives for key topics and giving several proofs of main results. For example, we
begin the analysis of random processes with several constructions of the Brownian

Preface ix
motion: 1) via binary subdivisions anticipating the later given Itô approach to con-
structing Markov evolutions, 2) via tightness of random-walk approximations, antici-
pating the later given LLN for non-homogeneous random walks, 3) via Hilbert-space
methods leading to Wiener chaos that is crucial for various developments, for instance
for Malliavin calculus and Feynman path integration, 4) via the Kolmogorov continu-
ity theorem. Similarly, we give two constructions of the Poisson process, several
constructions of basic stochastic integrals, several approaches to proving functional
CLTs (via tightness of random walks, Skorohod embedding and the analysis of gener-
ators). Further on various probabilistic and analytic constructions of the main classes
of Markov semigroups are given. Every effort was made to introduce all basic no-
tions in the most clear and transparent way, supplying intuition, developing examples
and stressing details and pitfalls that are crucial to grasp its full meaning in the general
context of stochastic analysis. Whenever possible, we opt for results with the simplest
meaningful formulation and quick direct proof.
As teaching and learning material, the book can be used on various levels and with
different objectives. For example, short courses on an introduction to Brownian mo-
tion, Lévy and Markov processes, or on probabilistic methods for PDE, can be based
on Chapters 2, 3 and 4 respectively, with chosen topics from other parts. Let us stress
only that the celebrated Itô’s lemma is not included in the monograph (it actually be-
came a common place in the textbooks). More advanced courses with various ﬂavors
can be built on part II, devoted, say, to continuous-time random walks, to probabilistic
methods for boundary value problems or for the Feynman path integral.
Finally, let me express my gratitude to Professor Niels Jacob from the University
of Wales, Swansea, an Editor of the De Gryuter Studies in Mathematics Series, who
encouraged me to write this book. I am also most grateful to Professor Nick Bingham
from Imperial College, London, for reading the manuscript carefully and making lots
of comments that helped me to improve the overall quality immensely.
Coventry, November 2010 V. N. Kolokoltsov

Notations
Numbers and sets
a _ b D max.a; b/, a ^ b D min.a; b/
N and Z are the sets of natural and integer numbers
Cd and Rd are the complex and real d-dimensional spaces, jxj or kxk for a
vector x 2 Rd denotes its Euclidean norm, .x; y/ or xy denotes the scalar
product of the vectors x; y 2 Rd
Re a and Im a are the real and imaginary part of a complex number a
Br .x/ (resp. Br ) is the ball of radius r centered at x (resp. at the origin)
Sd is the d-dimensional unit sphere in RdC1
RC (resp. N
RC) is the the set of positive (resp. non-negative) numbers
N
and @ are the closure and the boundary respectively of the subset in a
metric space
S is the set of all mappings S !
Œx is the integer part of a real number x
Functions
B.S/ (resp. C.S/ or Cb.S/) for a complete metric space .S; / (usually S D Rd,
.x; y/ D kx yk) is the Banach space of bounded Borel measurable (resp.
bounded continuous) functions on S equipped with the sup-norm
kf k D sup
x2S
jf .x/j
Cc.S/ C.S/ consists of functions with a compact support
CLip.S/ C.S/ consists of Lipschitz continuous functions f , i.e. jf .x/
f .y/j .x; y/ with a constant ; CLip.S/ is a Banach space under the norm
kf kLip D sup
x
jf .x/j C sup
x¤y
jf .x/ f .y/j=jx yj
C1.S/ C.S/ consists of f such that limx!1 f .x/ D 0, i.e. 8 9 a compact
set K W supx…K jf .x/j (it is a closed subspace of C.S/ if S is locally
compact)

xii Notations
Ck.Rd / or Ck
b
.Rd / (sometimes shortly Ck) is the Banach space of k times
continuously differentiable functions with bounded derivatives on Rd with the
norm being the sum of the sup-norms of the function itself and all its partial
derivative up to and including order k
Ck
Lip.Rd / is the subspace of Ck.Rd / with all derivative up to and including
order k being Lipschitz continuous; it is a Banach space equipped with the norm
kf kCk
Lip
D kf kCk1 C kf .k/
kLip
Ck
c .Rd / D Cc.Rd / Ck.Rd /
rf D .r1f; : : : ; rd f / D . @f
@x1
; : : : ; @f
@xd
/, f 2 C1.Rd /
Lp.; F ; /; p 1; is the usual Banach space of (the equivalence classes of)
measurable functions f on the measure space such that
kf kp D
Z
jf jp
.x/.dx/
1=p
1
L1.; F ; P/ is the Banach space of (the equivalence classes of) measurable
functions f on the measure space with a finite sup-norm
kf k D ess supx2 jf .x/j
S.Rd / D ¹f 2 C1.Rd / W 8k; l 2 N, jxjkrl f 2 C1.Rd /º is the Schwartz
space of rapidly deceasing functions
.f; g/ D
R
f .x/g.x/ dx denotes the scalar product for functions f; g on Rd or
on a general measure space
1M is the indicator function of a set M (equals one or zero according to whether
its argument is in M or otherwise)
sgn is the sign function taking values C1; 0; 1 for positive, vanishing and neg-
ative values of the argument respectively
f D O.g/ means jf j Cg for some constant C
f D o.g/n!1 ” limn!1.f=g/ D 0
Measures
M.S/ (resp. P.S/) is the set of finite (positive) Borel measures (resp. probability
measures) on a metric space S
Msigned.S/ defines the Banach space of finite signed Borel measures on a metric
space S
jj for a signed measure is its (positive) total variation measure
.f; / D
R
S f .x/.dx/ for f 2 C.S/, 2 M.S/

Notations xiii
Matrices and linear operators
AT is the transpose to a matrix A
Ker.A/, Sp.A/, tr.A/ are the kernel, spectrum and trace of the operator A
kAkB is the norm of the operator A in a Banach space B
kAkB!C is the norm of the operator A as a mapping between Banach spaces B
and C
C.Œ0; t; B/ is the Banach space of continuous functions on Œ0; t with values in
the Banach space B equipped with the sup-norm kf k D sups2Œ0;t kf .s/k
Probability
E and P deﬁne the expectation and probability, Ex; Px for x 2 S (respectively
E; P for 2 P.S/) are the expectation and probability with respect to an
S-valued process started at x (respectively with the initial distribution )

Standard abbreviations
a.s. almost sure
i.i.d. independent identically distributed
l.h.s. left-hand side
r.h.s. right-hand side
r.v. random variable
BM Brownian motion
CLT central limit theorem
CTRW continuous time random walk
LLN law of large numbers
ODE ordinary differential equation
OU Ornstein–Uhlenbeck (process)
PDE partial differential equation
PDO partial differential operator
‰DE pseudo differential equation
‰DO pseudo differential operator
SDE stochastic differential equation

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When he had lost his pipe, he swore,
Just a mild damn, and nothing more;
And once he cursed
The government; but then he reckoned
The Lord forgave him for the first,
And justified the second.
And he was temperate in all his ways,
Was John;
He never drank, but when Thanksgiving days
Came on;
Never in summer on a fishing trip
Would he allow the smell on board his ship;
Only in winter or in autumn,
When a cramp or something caught him,
Would he take it, for he prized it,
Not for its depraved abuses,
But for its discreeter uses,
As his Church had authorized it.
The sun had never shone
On a kinder man than John,
Nor upon
A better Christian than was John.
He was good to his dog, he was good to his cat,
And his love went out to his horse;
He loved the Lord and his Church, of course,
For righteous was he in thought and act;
And his neighbors knew, in addition to that,
He loved his wife, as a matter of fact.
Now, one fine day it occurred to John,
That his last great cramp was on;
For nothing that the doctor wrote
Could stop that rattle in his throat.
He had broken his back upon the oar,
He had dried his last boat-load of cod,

And nothing was left for John any more,
But to drift in his boat to the port of God.
Creatures of Another Country
I
THE BIRD OF PARADISE
Answer my riddle, will you? Nay,
Do not toss your head that way,
With such a ruffle of passion.
I merely asked you who was fleeced
To pay the jeweller and modiste
For this last word in fashion.
I have a right, if you only knew,
To put this delicate point to you—
Those sapphires dancing on your crest,
That cluster of rubies on your breast,
That necklace there, those pearls! The price?
Who paid it? Bird of Paradise!
And the only kind of reply that came
Out of that vision of tropical flame
Was that little ruffle of passion.
A tango of color from scarlet to green
Evolved as I watched the beauty preen
Her plumes in that maddening fashion.
So I left the Bird of the Garden to call,
This time, upon the Bird of the Hall;
For my temples beat with the throb of fire,
And I could not find in that land of Desire

A cooling wind, or water, or ice
To quench a fever in Paradise.
And the only answer I got in the Hall
Was a glance of repulse from the belle of the Ball,
With a little ruffle of passion;
Though I had a right to ask, I am sure,
Who sent that tiara for her coiffure,
And that latest corsage of fashion.
Not those the jewels I gave her to wear,
Not those the drops that hung from her ear;
And my fever burned like a thirst in Sahara,
When that osprey swung above the tiara,
And I knew no wind, nor water, nor ice
Might cool this hell in Paradise.
II
THE EPIGRAPHER
His head was like his lore—antique,
His face was thin and sallow-sick,
With god-like accent he could speak
Of Egypt's reeds or Babylon's brick
Or sheep-skin codes in Arabic.
To justify the ways divine,
He had travelled Southern Asia through—
Gezir down in Palestine,
Lagash, Ur and Eridu,
The banks of Nile and Tigris too.
And every occult Hebrew tale
He could expound with learned ease,
From Aaron's rod to Jonah's whale.

He had held the skull of Rameses—
The one who died from boils and fleas.
Could tell how—saving Israel's peace—
The mighty Gabriel of the Lord
Put sand within the axle-grease
Of Pharaoh's chariots; and his horde
O'erwhelmed with water, fire and sword.
And he had tried Behistun Rock,
That Persian peak, and nearly clomb it;
His head had suffered from the shock
Of somersaulting from its summit—
Nor had he quite recovered from it.
From that time onward to the end,
His mind had had a touch of gloom;
His hours with jars and coins he'd spend,
And ashes looted from a tomb,—
Within his spare and narrow room.
His day's work done, with the last rune
Of a Hammurabi fragment read,
He took some water spiced with prune
And soda, which imbibed, he said
A Syrian prayer, and went to bed.
* * * * * * * *
And thus he trod life's narrow way,—
His soul as peaceful as a river—
His understanding heart all day
Kept faithful to a stagnant liver.
L'ENVOI.
When at last his stomach went by default,
His graduate students bore him afar

To the East where the Dead Sea waters are,
And pickled his bones in Eternal Salt.
Ode to December, 1917
Was ever night so wild as this—this bleak December night!
Veiled in the sombre shroud that sepulchred the day;
Why thus bereft of heaven's beams, of moon and starry light,
Are all its ancient charms in sorrow laid away?
The year dies out with drifted leaves, with winds
and floods of rain,
Companions of the tempest with its brood of fears;
And voices far above us echo back the world's great pain,
In tongueless language inarticulate through tears.
Why passed with such inevitable speed
The eager splendor of the awakening spring?
So little did it seem to know or heed
Our outward cries, our hidden murmuring;
It shone upon us shyly for some reason,
Then flew into the summer's briefer season,
And found, amidst its roses fully blown,
A transient radiance fleeter than its own.
How sweet the flowers grew in the woods last May!
The trillium, splashed by sunlight, jauntily
Awoke to match the whiteness of its ray
With white of blood-root and anemone.
Within the stray leaves on the humid ground,
Beside the fallen trunks of trees, were found
Numerous hepaticas whose lilac hue
Seemed woven of heaven's purple and its blue,

And, near at hand, a running streamlet told
Of treasure hidden in the marigold.
A little while they stayed; how short the space!
We watched them as the hours went by,
We looked again, and saw them die—
Thus did they pass away; but in their place,
In meadow and in vale sprang up
The daisy and the buttercup;
Then on the creeping slopes of sunny hills,
By winding dales and tortuous rills,
Blue vervain rose to greet the sun,
Ere half the summer's race was run;
And in the fields and on the plains.
By forest paths, by country lanes,
By wayside and in garden plot,
The bluebell and forget-me-not;
And fair the bottle-gentian grew
Beside the wintergreen and rue.
And everywhere around us from the throats
Of joyous birds pealed forth ecstatic praise—
Glad hymns in which were heard no notes
Of dim unrest and troubled lays.
The heart had never taught them sorrows,
Regretful yesterdays nor morrows;
Each morning brought them its full boon of light,
And in return they gave their gift of song—
Free utterance that had no tale of wrong
Within the horizon of their life to right;
And when the evening drew to twilight close,
Fell the light mantle of their calm repose.
Fled are they all;
The flowers and the birds,
In vain we call,
With cries too dumb for words.

The fragrance and the music gone,
The fire of sunset, flush of dawn,
The waterlily in the lake,
The robin's love-song in the brake;
All these are fled and gone,
And with us now the night,
The wild December night.
Far, far away upon the seas
The billows tell their agonies;
The ocean in its frenzied roar
Lashes the ramparts of the shore;
The tempest with its shattering thunder
Drives the iron bulwarks under;
The furies, in their path advancing,
Are seen around the breakers dancing;
The sea-mews, blinded by the light
Of mast-head signals, flaring bright,
Are rent by blow of spar and sail
Within the clutches of the gale,
And sailors, drenched by salt and foam,
Yearn for the fireside of their home.
And thus upon the land
Earth's ravage is laid bare;
Slapped by the storm's fierce hand,
The wildcat and the bear
Lie huddled in the sand
That marks their common lair;
The trees in angry lurch
That grew beside each other—
The hemlock and the birch—
Now strive with one another,
In strangely human mood,
Born of unnatural feud.

Around the hoary mountain sides
The storm hurls its impetuous shock,
Is answered by the torrent's tides,
The iron echoes of the rock.
Gone are the woodland notes of spring,
The airs of summer's short-lived breath,
The autumn, too, has taken wing,
The year has rushed into its death.
Gone, like the memory of a dream,
A rainbow hovering o'er a stream;
And we, of nature's joys bereft,
Are with her deepening shadows left,
With grey upon the sea,
And driftwood on the reef,
With winter in the tree,
And death within the leaf.
Far, far away, across the distant deep,
Heaven's lightnings flash from out a darker scroll;
Midnight and darkness in wild chaos keep
A dawnless vigil, as slow thunders roll
Over a world upon whose face the storm
Breaks, and within the terrors of eclipse,
Fall the swift strokes of Death, clothed in the form
Of some dread angel of Apocalypse.
There rides a tempest heedless of the check
Of law, and with no mandate but its will,
Whose function lies alone in power to wreck,
That never hears the fiat, Peace, be still!
There, through deep, winding valleys that had known
The quiet haunts of peasants; through the green,
Sweet-tufted verdure that the spring had sown;
Through glens where only roe and fawn were seen
In peace; through plains where once the sunset's brush
Placed its soft crimson on the silent streams;
There, through that land that often loved the hush
Of evening and the tenderness of dreams,

Rolls now the bugle with its alien blast,
The cry of battle on the midnight air,
The fiery summons to earth's legions massed
Mid bayonets gleaming in the rocket's glare;
And streams that to the North Sea once had brought
The dawn's white silver and the sunset's gold,
Now pour such tides as Nature never wrought.
The ruddier treasures of a wealth untold.
O Nature! Thou that lovest life
In herb and brute and feathered kind,
Who leadest from the night's long strife
The morn with rays of promise lined;
Who bringest forth the vital glow
To bathe the trees in glorious light,
And bid the woodland flowers grow,
Clothed spotless in their raiment bright;
Who givest food to hart and hare
Upon the snowy mountain's crest,
And to the ravens everywhere,
The storm-proof covert of their nest;—
Hast thou within thy bounteous plan,
So rich and measureless and mild,
No boon wherewith to succour man,
Thy youngest, feeblest, blindest child?
Prostrate upon a formless field,
Bedewed with unavailing tears,
While the slow hours, faltering, yield
This nameless triad of the years;
What balm shall touch his stricken eyes?
What hand shall drive away his dead?
What tones shall quieten his cries?
What voice shall resurrect his dead?
O Winds; that sweep the surges from the bosom of the sea,
Strong with a strength unmeasured, as the chainless
lightnings—free;

Ye nether rivals of the thunders, as their voice your own,
Yet theirs excelling in your major harmonies of tone;
Ye mighty arbiters of light and shade, of hope and gloom,
Who fashion for the morn its cradle, for the eve its tomb,
Who garrison the towers of God with clouds in dark array,
Marshalling their watch and slumber till their hidden fires play;
All day ye played upon the forest pines a mournful strain,
As if the slowly ebbing year were laboring in its pain;
Upon the land ye tossed the agéd leaves in aimless quest,
And on the deep ye filled the sailor's heart with wild unrest.
O Winds! that stir the ashes of our altars while our cries
From hearthstone and from chancel in our agony arise,
That drive us in our frantic hours to prayer upon our knees,
While those we love drift shelterless upon the homeless seas;
O lift us once again to God! this time on kindlier wings—
So weary are we of the strife and fear the tempest brings;
Give us the vision of His gardens under skies of blue,
We have lived so long in shadow of the cypress and the yew;
Sing through the swell that crowns the ocean
when its rage has passed,
Resign the terrors of the gale, the furies of the blast;
Then through the vibrant music of the lyre of sea and land
Which our storm-sated world first heard when
from the Creator's hand
It rose at the Great Dawn, breathe soon that
sweet, untroubled peace,
That vista of life's cravings reared on hopes that never cease;
Blow out upon the raven plumes of this December night,
The world's unresting miseries, her shadow and her blight;
The story of her passions, and her dark, unfathomed sin,
The outward blow that slaughters, and the guilt
that slays within;
And deep from out the storm's last throes, peal
forth in life re-born,
The blazon of the future with the heralds of the morn;

The anthem of a world re-strung to human love and grace,
The full-toned orchestration of the heart-throbs of the race.
Newfoundland
Here the tides flow,
And here they ebb;
Not with that dull, unsinewed tread of waters
Held under bonds to move
Around unpeopled shores—
Moon-driven through a timeless circuit
Of invasion and retreat;
But with a lusty stroke of life
Pounding at stubborn gates,
That they might run
Within the sluices of men's hearts,
Leap under throb of pulse and nerve,
And teach the sea's strong voice
To learn the harmonies of new floods,
The peal of cataract,
And the soft wash of currents
Against resilient banks,
Or the broken rhythms from old chords
Along dark passages
That once were pathways of authentic fires
And swept by the wings of dream.
Red is the sea-kelp on the beach,
Red as the heart's blood,
Nor is there power in tide or sun
To bleach its stain.
It lies there piled thick
Above the gulch-line.

It is rooted in the joints of rocks,
It is tangled around a spar,
It covers a broken rudder,
It is red as the heart's blood,
And salt as tears.
Here the winds blow,
And here they die,
Not with that wild, exotic rage
That vainly sweeps untrodden shores,
But with familiar breath
Holding a partnership with life,
Resonant with the hopes of spring,
Pungent with the airs of harvest.
They call with the silver fifes of the sea,
They breathe with the lungs of men,
They are one with the tides of the sea,
They are one with the tides of the heart,
They blow with the rising octaves of dawn.
They die with the largo of dusk,
Their hands are full to the overflow,
In their right is the bread of life,
In their left are the waters of death.
Scattered on boom
And rudder and weed
Are tangles of shells;
Some with backs of crusted bronze,
And faces of porcelain blue,
Some crushed by the beach stones
To chips of jade;
And some are spiral-cleft
Spreading their tracery on the sand
In the rich veining of an agate's heart;
And others remain unscarred,
To babble of the passing of the winds.

Here the crags
Meet with winds and tides—
Not with that blind interchange
Of blow for blow
That spills the thunder of insentient seas;
But with the mind that reads assault
In crouch and leap and the quick stealth,
Stiffening the muscles of the waves.
Here they flank the harbors,
Keeping watch
On thresholds, altars and the fires of home,
Or, like mastiffs,
Over-zealous,
Guard too well.
Tide and wind and crag,
Sea-weed and sea-shell
And broken rudder—
And the story is told
Of human veins and pulses,
Of eternal pathways of fire,
Of dreams that survive the night,
Of doors held ajar in storms.
Flashlights and Echoes
From the Years of 1914 and 1915
I
A COAST

Scaling where a hundred crags
Disclose their high, precipitous walls,
Up hidden clefts and burnished jags,
The shore-line like a python crawls.
Along a league of ridges overspread
With the dead trunks of pine and oak, it drags
A roughening path; around the head
Of the last bluff it climbs, then falls,
Spilling its folds on spur and boulder,
Down a deep gulch where it rears and sprawls
Upon the Cape's lean shoulder.
Rolling dusks and vapors pour
A turgid silence on the shore,
Broken by a curlew screaming,
And a low, regurgitant note
Borne in from the laboring throat
Of a wave along a line of basalt streaming;
And, further off, where denser gloom
The headland and a reef-curve hides,
Falls the ground-swell's muttered boom
From the belfries of the tides.
Under a tattered curtain of fog
A flaw of wind makes the waters start;
They drift and scud and whirl;
And, held a moment near the heart
Of the eddy, a waterspout,—
Or some wild thing with twisted shape,
Compact of mist and wind and surge—
Hangs like a felon off the Cape.
II
LATER

(A man speaks)
Was that a cry you say you heard?
Where? No. The winds would drown it quite.
No sound would reach the shore to-night,
Except the scream of some wild bird.
A flash, you say, that cut the rain
Like a red knife? It could not be;
There's nothing living in this sea.
Don't look so frightened. What—again?
The lifeboat! They are hailing me.
They need a man for the stern oar;
The wind drives dead upon this shore,
A rudder's helpless in this sea.
III
(A woman speaks).
No. That was not a scream I heard;
One could not hear so far away.
That flash was but the breakers' spray,
That cry, the note of some wild bird.
IV
MORNING
I would not know him had I not
Once marked for him that tattoo spot—

A ship with flying-jib and spanker,
And underneath a chain and anchor.
Nor I, but for that reefer flap
Of moleskin, and this oilskin cap
I found a gunshot from the shore,
I'd know it from a hundred more.
We cannot take him home this way.
'Twould kill the woman straight to lay
The lad like this upon the bed,
And fetch her in to see him dead.
There is a chance she might not know
It was her son—he's battered so.
She'd know him by some canny trace,
Such as that birth-mark on his face,
And, what would smite her like a brand,
This stumped, third finger of his hand.
This coat and cap will tell her all;
We'll get him buried by night-fall;
There is no need to tell her more—
That we found the body on the shore.
V
GREAT TIDES
Great Tides! You filled the reaches
Under the North's wild blow;
Yet could not spare this smaller cup
Its salter overflow.

Huge hands! You rear our bulwarks up
With power to none akin;
Yet cannot lift a door-latch up
That a lad may enter in.
VI
THE AFTER-CALM
What is that color on the sea,
Dotted by the white sails of ships?
It is blue, you say. We know it not, and yet
We know the blue of violet,
The hue of mid-day skies,
And the sapphire of young children's eyes;
But that we do not know—unless it be
The pallor of dead lips.
That band upon the sea?
A sash of green that in a moment's time
Becomes a girdle of wrought gold,
Held by a silver clasp of surge.
It cannot be.
That green is now a belt of slime,
And now—an iron-knotted scourge,
And now—the form of some anguineal fold.
That crimson core with sepia fringe,
And orange tints between,
Shows how the sun's white alchemy
In vain attempt is seen
To paint a pansy on the sea.

That red is not the pansy's red,
Nor what the garden poppy shows,
Nor the vermilion that is spread
Upon the pastel of the rose,
But some deep smear that has its name
In the sprawled characters of the flood,
A splash of fire, a troubled flame,
That takes its color from the blood
Of one who through the night had died,
Breaking his body on the tide.
VII
SCENES FROM AFAR
(A Battlefield)
Above the tottering ramparts of the day
Massed clouds dissolve their lines; reform, and break
Into a thousand fragments from the grey.
Scattered, they drift awhile, then come to rest
On some far shore like mariners marooned,
While down the burning avenue of the west
The sun drops, flaming, like an angry wound.
A raven rises from the eastern skies,
Mounts up the lifted causeways of the north,
Winging an arc of shadow as she flies;
And soon the broken fragments close again,
The straylings of her brood flock to her wings—
Whirlwind and cloud, the thunder and the rain,
And what is left of night's unuttered things.
Now closed is every seam of sky and land.
The air, the water and the sod are one,

And every gulf of light and darkness spanned.
O spirits that love the daylight and the sun,
That with unerring fingers trace,
When night's dark moments are outrun,
The swarthy features of the morning's face;
In whose involvéd weavings hour by hour
Are fashioned forth the hues of nature's dress,
In dew and rainbow, grass and tree and flower,
And all the patterns of earth's loveliness;
Whose iridescent splendors burn
In vein of leaf, in curl of fern.
And in the flame the summer throws
Upon the poppy and the rose!
Draw near with every voice that's heard
In sound of cataract and bird,
With every color that the spring
Sheds on a blossom, blade or wing:
Come with your potencies that stir
The sap of life in pine and fir
That high along the mountains climb;
Bring rosemary and thorn and thyme
And heather—all that dawn distils
Of fragrance from your clouded hills:
From heath and glade and marge of lake,
Draw near and watch the morning break!
Wherefore should a daisy bloom,
Or scent come from the thorn?
What sun could penetrate this gloom,
Make redolent this morn?
The lark is banished from the sky,
The thrush has fled the ground,
Not heaven's chorus could outvie
This bacchanal of sound
That from the throat of fire and flood
Would drown the voice of God,

Answering the challenge of the blood
That cries out from the clod.
Where are the lilies that your valleys yield,
Or those that in foul waters blow?
May not the primrose of the field
Bloom near the snow?
Should not the clover in the meadows bare,
The sweet-briar in the hedges there,
Burst red and grow?
They cannot bloom. Spring's gales have lost
Their power the earth to leaven,
For those dark vapors would exhaust
The lavender of heaven.
VIII
A DIRGE
Now let the earth take
Into its care,
All that it travailed for,
All that it bare.
Leaves of the forest,
Yellow and red,
The drifting and scattered,
The dying and dead;
Grass of the hill-slopes,
Sickled and dried,
Vines that over-night
Blasted and died;

Blossoms and flowers
Nipped with the cold,
Trees that have fallen
A century old;
Moths of the candle-flame,
Gnats from the stream,
Wraiths from the moonlight,
Spectres of dream;
All that the earth gave,
All that it bare—
With all its far kindred
Of water and air.
And in those rutted acres
Which the heart's red blood has sown,
Soon shall the bramble flourish
Where the gentian had grown;
And wherever ran the myrtle,
Let the dust of thistles be shed,
For these, with nightshade and burdock,
Shall fast cover the dead.
IX
THE SEED MUST DIE
Ye meadows, groves, your birth renew; ye orchards, vineyards,
grow!
Where fast the wastrel waters of the Marne and Yser flow;
On the plains bestow your verdure, to the hills your odors fling.
Before the smile of Ceres, let your golden censer swing.

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Markov Processes Semigroups and Generators 1st Edition Vassili N. Kolokoltsov

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