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Diffusions, Superdiffusions and Partial Differential Equations
E. B. Dynkin, Cornell University, Ithaca, NY
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Colloquium Publications
2002; 236 pp; hardcover
Volume: 50
ISBN-10: 0-8218-3174-7
ISBN-13: 978-0-8218-3174-8
List Price: US$57 Member Price: US$45.60
Order Code: COLL/50

Interactions between the theory of partial differential equations of elliptic and parabolic types and the theory of stochastic processes are beneficial for both probability theory and analysis. At the beginning, mostly analytic results were used by probabilists. More recently, analysts (and physicists) took inspiration from the probabilistic approach. Of course, the development of analysis in general and of the theory of partial differential equations in particular, was motivated to a great extent by 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 differential equations and the corresponding Markov processes called diffusions and superdiffusions. 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.

Similar to the other books by Dynkin, Markov Processes (Springer-Verlag), Controlled Markov Processes (Springer-Verlag), and An Introduction to Branching Measure-Valued Processes (American Mathematical Society), this book can become a classical account of the presented topics.

Graduate students and research mathematicians interested in stochastic processes and partial differential equations.

Reviews

"This book makes a significant contribution to a field in which most of the main contributions since 1988 have been made by the author and Kuznetsov ... the organization of the book is well-thought-out, the presentation is systematic, and the key points are easy to understand ... the aim of the present book to build a bridge between superprocesses and semilinear differential equations is achieved nicely. This is nearly a miracle ... Dynkin's carefully written book, containing a pleasing depth of material within the topics presented should become a new fundamental reference ... because of the nature of the material and its presentation in a lively and engaging style, the book will indeed be accessible to graduate students and motivated readers with some basic knowledge of probability, functional analysis, and PDEs."

-- Mathematical Reviews

"The book begins with an excellently written introduction, in which the most important results are collected in concise and very easily understandable form ... For scientists and doctoral students, Dynkin's book is an altogether successful introduction to a fascinating and current research area on the frontiers of probability theory and analysis."

-- translated from Jahresbericht der Deutschen Mathematiker-Vereiningung

• Introduction
Parabolic equations and branching exit Markov systems
• Linear parabolic equations and diffusions
• Branching exit Markov systems
• Superprocesses
• Semilinear parabolic equations and superdiffusions
Elliptic equations and diffusions
• Linear elliptic equations and diffusions
• Positive harmonic functions
• Moderate solutions of $$Lu=\psi(u)$$
• Stochastic boundary values of solutions
• Rough trace
• Fine trace
• Martin capacity and classes $$\mathcal{N}_1$$ and $$\mathcal{N}_0$$
• Null sets and polar sets
• Survey of related results
• Basic facts of Markov processes and Martingales
• Facts on elliptic differential equations
• Epilogue
• Bibliography
• Subject index
• Notation index