Helge Holden, Bernt Øksendal, Jan Ubøe

Stochastic Partial Differential Equations

A Modeling, White Noise Functional Approach
Mitwirkender: Oksendal, Bernt; Uboe, Jan; Holden, Helge

• Broschiertes Buch

Produktbeschreibung

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 Lévy 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 Lévy processes.

The first part of the bookdeals with the classical Brownian motion case. The second extends it to the Lévy 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 book... It will be of great value for students ofprobability 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

Produktdetails

• Universitext
• Verlag: Springer / Springer New York / Springer, Berlin
• Artikelnr. des Verlages: 12517923, 978-0-387-89487-4
• 2nd 2010 edition
• Seitenzahl: 305
• Erscheinungstermin: 4. Dezember 2009
• Englisch
• Abmessung: 236mm x 158mm x 23mm
• Gewicht: 476g
• ISBN-13: 9780387894874
• ISBN-10: 038789487X
• Artikelnr.: 26167766

Autorenporträt

Helge Holden is a professor of mathematics at the Norwegian University of Science and Technology and an adjunt professor at the Center of Mathematics for Applications, part of the University of Oslo.  He has done extensive research in stochastic analysis, in particular in its application to flow in porous media. Bernt Øksendal is a professor at the Center of Mathematics for Applications at the University of Oslo.  He is a winner of the Nansen Prize for research in stochastic analysis and its applications. Jan Ubøe is a professor in the Department of Finance and Management Sciences at the Norwegian School of Economics and Business Administration.  He has written many papers about this subject. Tusheng Zhang is a professor of probability at the University of Manchester.  His current area of research is stochastic differential and partial differential equations, and he recently published a monograph on fractional Brownian fields with Bernt Øksendal and others.

Inhaltsangabe

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 Lévy 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 Lévy processes- References.- List of frequently used notation and symbols.- Index.

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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.

Rezensionen

"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 book... 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

"This book will be invaluable to anyone interested in doing research in white noise theory or in applying this theory to solving real-world problems." -Computing Reviews