Author | Holden, Helge. author |
---|---|
Title | Stochastic Partial Differential Equations [electronic resource] : A Modeling, White Noise Functional Approach / by Helge Holden, Bernt รksendal, Jan Ubรธe, Tusheng Zhang |
Imprint | Boston, MA : Birkhรคuser Boston, 1996 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-9215-6 |
Descript | XII, 231 p. online resource |
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 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