Table Of ContentDiffusions, superdiffusions and partial differential
equations
E. B. Dynkin
Department of Mathematics, Cornell University, Malott Hall,
Ithaca, New York, 14853
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 u˙ +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-familyto 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 limitsof 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 V 85
Q
5. Boundary value problems 88
6. Notes 91
3
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 ξ on the Martin boundary 109
ζ−
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 Z and Z 136
1 0
3. A relation between superdiffusions and conditionaldiffusions 138
4. Notes 140
Chapter 10. Rough trace 141
1. Definition and preliminary discussion 141
2. Characterization of traces 145
3. Solutions w with Borel B 147
B
4. Notes 151
Chapter 11. Fine trace 153
1. Singularity set SG(u) 153
2. Convexity properties of V 155
D
3. Functions J 156
u
4. Properties of SG(u) 160
5. Fine topology in E0 161
6. Auxiliary propositions 162
7. Fine trace 163
8. On solutions w 165
O
9. Notes 166
Chapter 12. Martin capacity and classes N and N 167
1 0
1. Martin capacity 167
2. Auxiliary propositions 168
3. Proof of the main theorem 170
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 divisibilityand 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
Preface
Interactions between the theory of partial differential equations of elliptic and
parabolictypesandthetheoryofstochasticprocesses arebeneficialfor,both,prob-
abilitytheory and analysis. At the beginning,mostlyanalyticresults were used by
probabilists. More recently, the analysts (andphysicists) took inspirationfrom 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 mathematicaltools
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 byasecondorderellipticdifferentialoperatorL. Aspecialcaseisthe
Brownian motion corresponding to the Laplacian ∆. A superdiffusion describes a
randomevolutionofacloudofparticles. Itis closelyrelated toequationsinvolving
an operator Lu−ψ(u). Here ψ belongs to a class of functions which contains, in
particularψ(u)=uαwithα>1. Fundamentalcontributionstotheanalytictheory
of equations
(0.1) Lu=ψ(u)
and
(0.2) u˙ +Lu=ψ(u)
weremadebyKeller,Osserman,Brezis andStrauss,LoewnerandNirenberg,Brezis
and V´eron, Baras and Pierre, Marcus and V´eron.
Arelationbetweentheequation(0.1)andsuperdiffusionswasestablished,first,
by S. Watanabe. Dawson and Perkins obtained deep results on the path behavior
of the super-Brownian motion. For applyinga superdiffusion to partialdifferential
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
usedtoinvestigateboundaryvalueproblemsforsemilinearequations. Inthepresent
book we deduce the entire theory of superdiffusion from these properties.
Weuse acombinationofprobabilisticandanalytictoolstoinvestigatepositive
solutions of equations (0.1)and (0.2). In particular, we study removable singulari-
tiesofsuch solutionsandacharacterizationofasolutionbyitstrace onthebound-
ary. These problems were investigated recently by a number of authors. Marcus
andV´eronusedpurelyanalyticmethods. LeGall,DynkinandKuznetsovcombined
7
8 PREFACE
probabilisticand analyticapproach. Le Gallinvented 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,withsimplifiedproofs)oftheresultsobtainedsince1988inaseriesofpapers
of Dynkin and Dynkin and Kuznetsov. Many results obtained originallyby using
superdiffusions are extended in the book to more general equations by applying a
combinationofdiffusionswithpurelyanalyticmethods. Almostallchaptersinvolve
a mixture of probabilityand analysis. Exceptions are Chapters 7 and 9 where the
probabilityprevailsandChapter 13where itisabsent. Independently oftherest of
thebook,Chapter7canserveasanintroductiontotheMartinboundarytheoryfor
diffusionsbased on Hunt’sideas. A contributiontothe theory ofMarkovprocesses
isalso anew formofthe strong Markovproperty inatime inhomogeneoussetting.
The theory of parabolic partial differential equations has a lot of similarities
withthetheoryofellipticequations. Manyresultsonellipticequationscanbeeasily
deduced from the results on parabolic equations. On the other hand, the analytic
technique needed inthe parabolicsetting ismore complicatedand the mostresults
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 (includingsuperdiffusions) are treated as aspecial case ofsuch 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.)
InPart2wedealwithellipticdifferentialequationsandwithtime-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
singularitiesofsuchsolutions(both,fromanalyticandprobabilisticpointsofview).
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.
Chapter1is aninformalintroductionwhere we present someofthe basicideas
and tools used in the rest of the book. We consider an elliptic setting and, to
simplifythe presentation,we restrict ourselves toaparticularcase oftheLaplacian
∆ (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].
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.
IamgratefultoS.E. Kuznetsov for manydiscussions whichleadto the clarifi-
cationofanumberofpointsinthepresentation. Iamindebtedtohimforproviding
mehisnotesonrelationsbetweenremovableboundarysingularitiesandthePoisson
capacity. (They were used in the work on Chapter 13.) I am also indebted to P.
J.Fitzsimmons forthe notes on hisapproach to the construction ofsuperprocesses
(used in Chapter 4) and to J.-F. Le Gallwhose comments helped to fillsome gaps
in the expository part of the book.
I take this opportunity to thank experts on PDEs who gladly advised me on
the literatureintheir field. Especiallyimportantwastheassistance ofN.V.Krylov
and V. G. Maz’ya.
MyforemostthanksgotoYuan-chungSheuwhoreadcarefullytheentire man-
uscript and suggested numerous corrections and improvements.
The research of the author connected with this volume was supported in part
by the NationalScience FoundationGrant DMS -9970942.
