Stochastic partial differential equations : a modeling, white noise functional approach

Author(s)

• Holden, H. (Helge)

Bibliographic Information

Stochastic partial differential equations : a modeling, white noise functional approach

Helge Holden ... [et al.]

（Probability and its applications）

Birkhäuser, 1996

• : sz
• : us

Note

Includes bibliographical references and index

Description and Table of Contents

Volume

: us ISBN 9780817639280

Description

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.

Table of Contents

• 1. Introduction.- 1.1. Modeling by stochastic differential equations.- 2. Framework.- 2.1. White noise.- The 1-dimensional, d-parameter smoothed white noise.- The (smoothed) white noise vector.- 2.2. The Wiener-Ito chaos expansion.- Chaos expansion in terms of Hermite polynomials.- Chaos expansion in terms of multiple Ito integrals.- 2.3. Stochastic test functions and stochastic distributions.- The Kondratiev spaces (S)pm
• N, (S)-pm
• N.- The Hida test function space(S) and the Hida distribution space(S)*.- Singular white noise.- 2.4. The Wick product.- Son e examples and counterexamples.- 2.5. Wick multiplication and Ito/Skorohod integration.- 2.6. The Hermite transform.- Tht characterization theorem for(S)-1N.- Positive noise.- The positive noise matrix.- 2.7. The S)p,rN spaces and the S-transform.- The S-transform.- 2.8. The topology of (S)-1N.- Stochastic distribution processes.- 2.9. The F-transform and the Wick product on L1 (?).- Functional processes.- 2.10. The Wick product and translation.- 2.11. Positivity.- Exercises.- 3. Applications to stochastic ordinary differential equations.- 3.1. Linear equations.- Linear 1-dimensional equations.- Some remarks on numerical simulations.- Some linear multi-dimensional equations.- 3.2. A model for population growth in a crowded stochastic environment.- The general(S)-1 solution.- A solution in L1(?).- A comparison of Model A and Model B.- 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.- Exercises.- 4. Stochastic partial differential equations.- 4.1. General remarks.- 4.2. The stochastic Poisson equation.- The functional process approach.- 4.3. The stochastic transport equation.- Pollution in a turbulent medium.- The heat equation with a stochastic potential.- 4.4. The stochastic Schroedinger equation.- L1 (?)-properties of the solution.- 4.5. The viscous Burgers' equation with a stochastic source.- 4.6. The stochastic pressure equation.- The smoothed positive noise case.- An inductive approximation procedure.- The 1-dimensional case.- The singular positive case.- 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.- Exercises.- Appendix A. The Bochner-Minlos theorem.- Appendix B. A brief review of Ito calculus.- The Ito 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.

Volume

: sz ISBN 9783764339289

Description

The main emphasis of this work is on stochastic partial differential equations. First the stochastic Poisson equation and the stochastic transport equation are discussed; then the authors go on to deal with the Schrodinger equation, the heat equation, the nonlinear Burgers' equation with a stochastic source, and the pressure equation. The white noise approach often allows for solutions given by explicit formulas in terms of expectations of certain auxiliary processes. The noise in the above examples are all of a Gaussian white noise type. In the end, the authors also show how to adapt the analysis to SPDEs involving noise of Poissonian type.

Table of Contents

• Introduction - modelling by stochastic differential equations. Framework - white noise: the 1-dimensional, d-parameter (smoothed) white noise, the (smoothed) white noise vector
• the Wiener-Ito chaos expansion: expansion in terms of hermite polynomials, chaos expansion in terms of multiple Ito integrals
• the Wick product: some examples and counterexamples
• Wick multiplication and Ito/Skorohod integration
• the Hermite transform: definition, relation to wick product
• the Wick product and translation
• positivity. Applications to stochastic ordinary differential equations - linear equations: linear 1-dimensional equations, some linear multi-dimensional equations
• a model for population growth in a crowded, stochastic environment
• a general existence and uniqueness theorem: the general linear multi-dimensional equation
• the stochastic Volterra equation
• Wick product versus ordinary product: a comparison experiment
• solution and Wick approximation of quasilinear SDE. Stochastic partial differential equations - general remarks
• the stochastic Poisson equation: the functional process approach
• the stochastic transport equation: pollution in a turbulent medium, the heat equation with a stochastic potential
• the viscous Burgers equation with a stochastic source
• the stochastic pressure equation: the smoothed positive noise case, an inductive approximation procedure, the 1-dimension case, the singular positive noise case
• the heat equation in a stochastic anisotropic medium
• a class of quasilinear parabolic SPDEs
• SPDEs driven by Poissonian noise. (Part contents).Poissonian noise. (Part contents).

by "Nielsen BookData"

Details

• NCID: BA28606321
• ISBN:
  • 3764339284
  • 0817639284
• LCCN: 96022105
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Boston, Mass.
• Pages/Volumes: 230 p.
• Size: 24 cm
• Classification:
  • LCC : QA274.25
  • DC20 : 519.2
  • NDC9 : 417.1
  • NDC8 : 417.1
• Subject Headings:
  • LCSH : Stochastic partial differential equations
• Parent Bibliography ID:
  • BA10668227