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Stochastic Partial Differential Equations

By Pao-Liu Chow

Series Editor: Goong Chen, Thomas J. Bridges

Published March 19th 2007 by Chapman and Hall/CRC – 281 pages

Series: Chapman & Hall/CRC Applied Mathematics & Nonlinear Science

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Description

As a relatively new area in mathematics, stochastic partial differential equations (PDEs) are still at a tender age and have not yet received much attention in the mathematical community. Filling the void of an introductory text in the field, Stochastic Partial Differential Equations introduces PDEs to students familiar with basic probability theory and Itô's equations, highlighting several computational and analytical techniques.

Without assuming specific knowledge of PDEs, the text includes many challenging problems in stochastic analysis and treats stochastic PDEs in a practical way. The author first brings the subject back to its root in classical concrete problems. He then discusses a unified theory of stochastic evolution equations and describes a few applied problems, including the random vibration of a nonlinear elastic beam and invariant measures for stochastic Navier-Stokes equations. The book concludes by pointing out the connection of stochastic PDEs to infinite-dimensional stochastic analysis.

By thoroughly covering the concepts and applications of stochastic PDEs at an introductory level, this text provides a guide to current research topics and lays the groundwork for further study.

Reviews

"The book provides an excellent introduction to the theory of Stochastic Partial Differential Equations . . . It provides a well written and timely contribution to the literature."

– Evelyn Buckwar, in Zentralblatt Math, 2009

"The text may be characterized as an excellent guide to current research topics that opens possibilities for further developments in the field."

– In EMS Newsletter, 2008

"This introductory book fills a gap in the field."

– Nikita Y. Ratanov, in Mathematical Reviews, 2008d

". . . very well written introductory book . . . Overall I thoroughly recommend this book and believe that it will be a useful textbook with which to introduce students and young scientists to computational and analytical techniques for stochastic differential equations. This book is of great interest to applied mathematicians, theoretical physicists, naturalists, and all interested in statistical formulation of scientific problems."

– Andrzej Icha, Institute of Mathematics, in Pure and Applied Geophysics, June 2005

Contents

PREFACE

PRELIMINARIES

Introduction

Some Examples

Brownian Motions and Martingales

Stochastic Integrals

Stochastic Differential Equations

Comments

SCALAR EQUATIONS OF FIRST ORDER

Introduction

Generalized Itô's Formula

Linear Stochastic Equations

Quasilinear Equations

General Remarks

STOCHASTIC PARABOLIC EQUATIONS

Introduction

Preliminaries

Solution of Random Heat Equation

Linear Equations with Additive Noise

Some Regularity Properties

Random Reaction-Diffusion Equations

Parabolic Equations with Gradient-Dependent Noise

STOCHASTIC PARABOLIC EQUATIONS IN THE WHOLE SPACE

Introduction

Preliminaries

Linear and Similinear Equations

Feynman-Kac Formula

Positivity of Solutions

Correlation Functions of Solutions

STOCHASTIC HYPERBOLIC EQUATIONS

Introduction

Preliminaries

Wave Equation with Additive Noise

Semilinear Wave Equations

Wave Equations in Unbounded Domain

Randomly Perturbed Hyperbolic Systems

STOCHASTIC EVOLUTION EQUATIONS IN HILBERT SPACES

Introduction

Hilbert Space-Valued Martingales

Stochastic Integrals in Hilbert Spaces

Itô's Formula

Stochastic Evolution Equations

Mild Solutions

Strong Solutions

Stochastic Evolution Equations of Second Order

ASYMPTOTIC BEHAVIOR OF SOLUTIONS

Introduction

Itô's Formula and Lyapunov Functionals

Boundedness of Solutions

Stability of Null Solution

Invariant Measures

Small Random Perturbation Problems

Large Deviations Problems

FURTHER APPLICATIONS

Introduction

Stochastic Burgers and Related Equations

Random Schrödinger Equation

Nonlinear Stochastic Beam Equations

Stochastic Stability of Cahn-Hilliard Equation

Invariant Measures for Stochastic Navier-Stokes Equations

DIFFUSION EQUATIONS IN INFINITE DIMENSIONS

Introduction

Diffusion Processes and Kolmogorov Equations

Gauss-Sobolev Spaces

Ornstein-Uhlenbeck Semigroup

Parabolic Equations and Related Elliptic Problems

Characteristic Functionals and Hopf Equations

REFERENCES

INDEX

Name: Stochastic Partial Differential Equations (Hardback) – Chapman and Hall/CRC 
Description: By Pao-Liu ChowSeries Editor: Goong Chen, Thomas J. Bridges. As a relatively new area in mathematics, stochastic partial differential equations (PDEs) are still at a tender age and have not yet received much attention in the mathematical community. Filling the void of an introductory text in the field, Stochastic...
Categories: Differential Equations, Mathematical Physics, Probability Theory & Applications