Stochastic Partial Differential Equations, Space-time White Noise and Random Fields

Abstract

This book is an introduction to the theory of stochastic partial differential equations (SPDEs), using the random field approach pioneered by J.B. Walsh (1986). It consists of two blocks: the core matter (Chapters 1 to 6) and the appendices (A to C). Chapter 1 introduces the subject, with a discussion of isonormal Gaussian processes, space-time white noise, and motivating examples of SPDEs. Chapter 2 presents a theory of stochastic integration with respect to space-time white noise. Chapter 3 deals with SPDEs with additive noise. In Chapter 4, we study a general class of SPDEs, in which additive and multiplicative nonlinearities appear. Chapter 5 discusses asymptotic properties of the solution to the stochastic heat equation such as existence of invariant and reversible measures, convergence in law to the invariant distribution, mixing and irreducibility. In Chapter 6, we prove a theorem on existence and uniqueness of solutions in the weak sense. Then we present a selection of important topics in the theory of SPDEs: the Markov field property, asymptotic bounds on moments of solutions that are useful for studying long-time behavior of the solutions, a comparison theorem for the stochastic heat equation, an introduction to potential theory for SPDEs, and a study of SPDEs with rough initial conditions. Appendix A summarises the main results from the theory of stochastic processes and stochastic analysis that are used throughout the book. Appendix B is devoted to a systematic presentation of properties of fundamental solutions and Green's functions associated to the classical linear differential operators (heat, fractional heat and wave operators). Appendix C is a toolbox section. Each chapter is followed by a "Notes" section, which gives historically important references, original sources and points towards other related important contributions.