Stochastic Partial Differential Equations [electronic resource] : A Modeling, White Noise Functional Approach / by Helge Holden, Bernt ksendal, Jan Ubȹe, Tusheng Zhang.

Preface to the Second Edition -- Preface to the First Edition -- Introduction -- Framework -- Applications to stochastic ordinary differential equations -- Stochastic partial differential equations driven by Brownian white noise -- Stochastic partial differential equations driven by LȨvy white noise -- Appendix A. The Bochner-Minlos theorem -- Appendix B. Stochastic calculus based on Brownian motion -- Appendix C. Properties of Hermite polynomials -- Appendix D. Independence of bases in Wick products -- Appendix E. Stochastic calculus based on LȨvy processes- References -- List of frequently used notation and symbols -- Index.

The first edition of Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, gave a comprehensive introduction to SPDEs driven by space-time Brownian motion noise. In this, the second edition, the authors extend the theory to include SPDEs driven by space-time LȨvy process noise, and introduce new applications of the field. Because the authors allow the noise to be in both space and time, the solutions to SPDEs are usually of the distribution type, rather than a classical random field. To make this study rigorous and as general as possible, the discussion of SPDEs is therefore placed in the context of Hida white noise theory. The key connection between white noise theory and SPDEs is that integration with respect to Brownian random fields can be expressed as integration with respect to the Lebesgue measure of the Wick product of the integrand with Brownian white noise, and similarly with LȨvy processes. The first part of the book deals with the classical Brownian motion case. The second extends it to the LȨvy white noise case. For SPDEs of the Wick type, a general solution method is given by means of the Hermite transform, which turns a given SPDE into a parameterized family of deterministic PDEs. Applications of this theory are emphasized throughout. The stochastic pressure equation for fluid flow in porous media is treated, as are applications to finance. Graduate students in pure and applied mathematics as well as researchers in SPDEs, physics, and engineering will find this introduction indispensible. Useful exercises are collected at the end of each chapter. From the reviews of the first edition: "The authors have made significant contributions to each of the areas. As a whole, the book is well organized and very carefully written and the details of the proofs are basically spelled out... This is a rich and demanding bookǪ It will be of great value for students of probability theory or SPDEs with an interest in the subject, and also for professional probabilists." Mathematical Reviews "...a comprehensive introduction to stochastic partial differential equations." Zentralblatt MATH

ZDB-2-SMA