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AuthorHolden, Helge. author
TitleStochastic Partial Differential Equations [electronic resource] : A Modeling, White Noise Functional Approach / by Helge Holden, Bernt ร{152}ksendal, Jan Ubรธe, Tusheng Zhang
ImprintBoston, MA : Birkhรคuser Boston, 1996
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Descript XII, 231 p. online resource


This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initiated by a group including some of us with the support of VISTA, a research cooperaยญ tion between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap A.S. (Statoil). The purpose of the project was to use stochastic partial differential equations (SPDEs) to describe the flow of fluid in a medium where some of the parameters, e.g., the permeability, were stochastic or "noisy". We soon realized that the theory of SPDEs at the time was insufficient to handle such equations. Therefore it became our aim to develop a new mathematically rigorous theory that satisfied the following conditions. 1) The theory should be physically meaningful and realistic, and the correยญ sponding solutions should make sense physically and should be useful in applications. 2) The theory should be general enough to handle many of the interesting SPDEs that occur in reservoir theory and related areas. 3) The theory should be strong and efficient enough to allow us to solve th,̃se SPDEs explicitly, or at least provide algorithms or approximations for the solutions


1. Introduction -- 1.1. Modeling by stochastic differential equations -- 2. Framework -- 2.1. White noise -- 2.2. The Wiener-Itรด chaos expansion -- 2.3. Stochastic test functions and stochastic distributions -- 2.4. The Wick product -- 2.5. Wick multiplication and Itรด/Skorohod integration -- 2.6. The Hermite transform -- 2.7. The S)p,rN spaces and the S-transform -- 2.8. The topology of (S)-1N -- 2.9. The F-transform and the Wick product on L1 (?) -- 2.10. The Wick product and translation -- 2.11. Positivity -- 3. Applications to stochastic ordinary differential equations -- 3.1. Linear equations -- 3.2. A model for population growth in a crowded stochastic environment -- 3.3. A general existence and uniqueness theorem -- 3.4. The stochastic Volterra equation -- 3.5. Wick products versus ordinary products: A comparison experiment Variance properties -- 3.6. Solution and Wick approximation of quasilinear SDE -- 4. Stochastic partial differential equations -- 4.1. General remarks -- 4.2. The stochastic Poisson equation -- 4.3. The stochastic transport equation -- 4.4. The stochastic Schrรถdinger equation -- 4.5. The viscous Burgersโ{128}{153} equation with a stochastic source -- 4.6. The stochastic pressure equation -- 4.7. The heat equation in a stochastic, anisotropic medium -- 4.8. A class of quasilinear parabolic SPDEs -- 4.9. SPDEs driven by Poissonian noise -- Appendix A. The Bochner-Minlos theorem -- Appendix B. A brief review of Itรด calculus -- The Itรด formula -- Stochastic differential equations -- The Girsanov theorem -- Appendix C. Properties of Hermite polynomials -- Appendix D. Independence of bases in Wick products -- References -- List of frequently used notation and symbols

Mathematics Partial differential equations Probabilities Physics Mathematics Probability Theory and Stochastic Processes Partial Differential Equations Theoretical Mathematical and Computational Physics


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