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Stochastic partial differential equations. (English) Zbl 1134.60043
Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. Boca Raton, FL: Chapman & Hall/CRC (ISBN 978-1-58488-443-9/hbk). ix, 281 p. (2007).
The book provides an excellent introduction to the theory of Stochastic Partial Differential Equations (SPDEs). The first chapter contains preliminary material concerning Brownian motion, martingales, stochastic integrals and stochastic differential equations. Here also some first examples of SPDEs, e.g., the stochastic sine-Gordon equation and the stochastic Burgers equation, are presented and a brief idea of the underlying application problem is given. In the next four chapters of the book some prototypes of SPDEs are studied, that is scalar equations of first order, stochastic parabolic equations and stochastic hyperbolic equations, where particular examples treated are stochastic heat and wave equations. The author analyses existence, uniqueness and regularity properties of their solutions constructively by applying the methods of eigenfunction expansions, the Green’s functions and Fourier transforms, together with techniques from stochastic analysis. In Chapters six to nine SPDEs are investigated based on the theory of stochastic evolution equations in Hilbert spaces. In particular, Chapter 6 provides background material on this topic, while Chapter 7 is concerned with issues of asymptotic behaviour of solutions of SPDEs, e.g., boundedness, stability, invariant measures and large deviation principles. The methods discussed are employed to study some examples arising in applications in Chapter 8. The last chapter briefly considers the connection between SPDEs and Markov diffusion processes in Hilbert spaces.
The book is intended for readers familiar with stochastic analysis, but not necessarily with the theory of partial differential equations. It provides a well written and timely contribution to the literature.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
35R60 PDEs with randomness, stochastic partial differential equations