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Stochastic Partial Differential Equations: An Introduction

Wei Liu and Michael Röckner
Publication Date: 
Number of Pages: 
[Reviewed by
Richard Durrett
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Stochastic (ordinary) differential equations, or SDEs, appear in a number of different contexts that range from population genetics to models of stock prices. These processes are driven by Brownian motions, so their paths are nondifferentiable and it takes some sophistication to understand the associated stochastic calculus. When one considers stochastic partial differential equations, however, the difficulties are magnified 10-fold. For example, unless the noise is very tame, function valued solutions exist only in one dimension. In higher dimensions one must deal with solutions that are generalized functions and one must confront technical problems such as: what is the meaning of a nonlinear function (e.g., a power) of such a solution.

Despite the technical difficulties there are many natural examples of SPDE, and several are covered in this book. For example, stochastic versions of Burgers equation, 2D and 3D Navier-Stokes, Cahn-Hilliard equations and surface growth models. A more recent example mentioned in the introductory chapter is the KPZ equation. Martin Hairer won the Fields Medal in 2014 for his work proving existence of solutions (and other contributions).

There are several different approaches to SPDE. The martingale approach, which dates back to John Walsh's 1986 St. Flour notes, the semigroup or mild solution approach developed in two books by da Prato and Zabczy, and the “variational approach,” considered here. The volume contains a complete presentation of the main existence and uniqueness results in the case of locally monotone coefficients. The extension of the “well-known” case of globally monotone coefficients substantially widens the applicability of the results, leads to a unified approach and to simplified proofs in many classical examples.

To keep the book self-contained, necessary results about SDEs in finite dimensions are also included with complete proofs, as well as a chapter on stochastic integration on Hilbert spaces. Further fundamentals are treated in a nine-part appendix. The previous version of this book grew out of a year-long graduate course at Purdue in 2005–2006; the current version out of a course at Bielefeld in 2012–2013. However, it would most likely be used as for a one semester course for graduate students who have had a course in measure-theoretic probability including martingales and the basics of stochastic processes. The authors state that a knowledge of stochastic integration is not formally required, but I think a student without that background would rapidly get lost.

I like that (i) the book starts with eight significant examples on pages 5–6 which it returns to as the theory is developed, and (ii) as it develops the theory it keeps the reader focused on the main results. Obviously a book that starts with a chapter on stochastic integration in Hilbert space is not for undergraduates. However, graduate students and researchers who are interested in this area will find it a clear introduction.

Richard Durrett taught at UCLA and Cornell before he came to Duke in 2010. He is a member of the National Academy of Science, who for the last thirty years has used probability to study problems that arise from ecology, genetics, and cancer modeling.

See the table of contents in the publisher's webpage.