Stochastic Partial Differential Equations

A Modeling, White Noise Functional Approach

Helge Holden, Bernt Oksendal, Jan Uboe, Tusheng Zhang (Autoren)

Buch | Softcover

231 Seiten

2012 | Softcover reprint of the original 1st ed. 1996
Birkhauser Boston Inc (Verlag)
978-1-4684-9217-0 (ISBN)

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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 coopera 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 corre 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-Itô chaos expansion.- 2.3. Stochastic test functions and stochastic distributions.- 2.4. The Wick product.- 2.5. Wick multiplication and Itô/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 Schrödinger equation.- 4.5. The viscous Burgers’ 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 Itô calculus.- The Itô 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.

Erscheint lt. Verlag	16.6.2012
Reihe/Serie	Probability and Its Applications
Zusatzinfo	XII, 231 p.
Verlagsort	Secaucus
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
	Naturwissenschaften ► Physik / Astronomie
ISBN-10	1-4684-9217-9 / 1468492179
ISBN-13	978-1-4684-9217-0 / 9781468492170
Zustand	Neuware