Stochastic Partial Differential Equations

This book gives a concise introduction to the classical theory of stochastic partial differential equations (SPDEs). It begins by describing the classes of equations which are studied later in the book, together with a list of motivating examples of SPDEs which are used in physics, population dynamics, neurophysiology, finance and signal processing. The central part of the book studies SPDEs as infinite-dimensional SDEs, based on the variational approach to PDEs. This extends both the classical Itô formulation and the martingale problem approach due to Stroock and Varadhan. The final chapter considers the solution of a space-time white noise-driven SPDE as a real-valued function of time and (one-dimensional) space. The results of J. Walsh's St Flour notes on the existence, uniqueness and Hölder regularity of the solution are presented. In addition, conditions are given under which the solution remains nonnegative, and the Malliavin calculus is applied. Lastly, reflected SPDEs and their connection with super Brownian motion are considered.At a time when new sophisticated branches of the subject are being developed, this book will be a welcome reference on classical SPDEs for newcomers to the theory.

“This monograph is a nice addition to the existing literature on SPDEs, complementing it by presenting a variety of variational methods available for nonlinear SPDEs.” (Kristin Kirchner, SIAM Review, Vol. 65 (1), March, 2023)“This book … serves as a solid introduction to recent developments in the classical theory of stochastic partial differential equations. This book is a nice introduction to the subject matter for those unfamiliar with the topic. The author is direct, and the small size of the book makes it very approachable. Plenty of references are provided for those who wish to dive deeper into the topic.” (Bill Satzer, MAA Reviews, June 6, 2022)

Etienne Pardoux is professor emeritus at the Institut de Mathématiques de Marseille, within Aix Marseille Univ. His research has covered several topics of stochastic analysis, in particular stochastic partial differential equations, backward stochastic differential equations and homogenization. More recently, he has turned his interests towards evolutionary biology and modeling of infectious diseases. He is the author of more than 160 publications, including four books.