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A concise course on stochastic partial differential equations. (English) Zbl 1123.60001
Lecture Notes in Mathematics 1905. Berlin: Springer (ISBN 978-3-540-70780-6/pbk; 978-3-540-70781-3/ebook). vi, 144 p. (2007).
As the title promises these 150 page lecture notes provide “A Concise Course on Stochastic Partial Differential Equations”, more precisely, an introdution to nonlinear stochastic partial differential equations (SPDEs) of evolutionary type with respect to a given cylindrical Wiener process. The amount of material comprises somewhat more than a one semester course and requires knownledge of an advanced course in probability theory and basics of functional analysis. After a short motivation in Chapter 1, the following section is entirely dedicated to the construction of Wiener processes and cylindrical Wiener processes on Hilbert spaces as well as to their integration theory, respectively. In Chapter 3 an existence and uniqueness result for stochastic differential equations in finite dimensions is worked out in detail under very general local weak monotonicity and coercivity conditions on the coefficients. After these propaedeutic expositions the authors come to the heart of this book, the investigation of a class of stochastic partial differential equations, that covers most of the motivating examples like nonlinear reaction diffusion equations and with particular emphasis the porous medium equation. In a first step the authors discuss the ensemble of conditions on the coefficients, followed by the statement of an Itô formula and its application in order to proof the existence and uniqueness of variational solutions for the previously mentioned class of SPDEs. The closure of this account is the proof of the Markov property and existence of an invariant measure. This monograph is an elegantly and economically written first introduction to the field and meets the expectations of the title entirely. A great advantage of this account is its wide self-containance of the plot, the completeness of all proofs, as well as a nice overview over the different notions of solutions of SPDEs culminating in the Yamada-Watanabe theorem entirely proven in the appendix. This book might be particularly helpful for graduate students and young researchers to get acquainted with this sophisticated area of research.

MSC:
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60Hxx Stochastic analysis
35R60 PDEs with randomness, stochastic partial differential equations
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