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Diffusions, Superdiffusions and Partial Differential Equations E. B. Dynkin

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Diffusions, Superdiffusions and Partial Differential Equations E. B. Dynkin Diffusions, 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 equationu ˙ + 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 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 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 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)u ˙ + Lu = ψ(u) were made by Keller, Osserman, Brezis and Strauss, Loewner and Nirenberg, Brezis and V´eron,Baras and Pierre, Marcus and V´eron. 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 V´eron used purely analytic methods. Le Gall, Dynkin and Kuznetsov combined 7 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.
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