Stochastic Partial Differential Equations: 2nd Edition (Hardback) book cover

Stochastic Partial Differential Equations

2nd Edition

By Pao-Liu Chow

Chapman and Hall/CRC

334 pages

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pub: 2014-12-10
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Description

Explore Theory and Techniques to Solve Physical, Biological, and Financial Problems

Since the first edition was published, there has been a surge of interest in stochastic partial differential equations (PDEs) driven by the Lévy type of noise. Stochastic Partial Differential Equations, Second Edition incorporates these recent developments and improves the presentation of material.

New to the Second Edition

  • Two sections on the Lévy type of stochastic integrals and the related stochastic differential equations in finite dimensions
  • Discussions of Poisson random fields and related stochastic integrals, the solution of a stochastic heat equation with Poisson noise, and mild solutions to linear and nonlinear parabolic equations with Poisson noises
  • Two sections on linear and semilinear wave equations driven by the Poisson type of noises
  • Treatment of the Poisson stochastic integral in a Hilbert space and mild solutions of stochastic evolutions with Poisson noises
  • Revised proofs and new theorems, such as explosive solutions of stochastic reaction diffusion equations
  • Additional applications of stochastic PDEs to population biology and finance
  • Updated section on parabolic equations and related elliptic problems in Gauss–Sobolev spaces

The book covers basic theory as well as computational and analytical techniques to solve physical, biological, and financial problems. It first presents classical concrete problems before proceeding to a unified theory of stochastic evolution equations and describing applications, such as turbulence in fluid dynamics, a spatial population growth model in a random environment, and a stochastic model in bond market theory. The author also explores the connection of stochastic PDEs to infinite-dimensional stochastic analysis.

Reviews

"This is the second edition of the very well-written and introductory, application-oriented book on stochastic partial differential equations (SPDEs) by P.L. Chow. Compared to the first edition, the main change is adding new materials about SPDEs driven by Lévy-type noise."

Zentralblatt MATH 1321

Praise for the First Edition:

"The book provides an excellent introduction to the theory of stochastic partial differential equations … a well-written and timely contribution to the literature."

—Evelyn Buckwar, Zentralblatt Math, 2009

"… an excellent guide to current research topics that opens possibilities for further developments in the field."

EMS Newsletter, 2008

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

—Nikita Y. Ratanov, Mathematical Reviews, 2008d

"… very well-written introductory book … 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 the statistical formulation of scientific problems."

—Andrzej Icha, Pure and Applied Geophysics, June 2005

Table of Contents

Preliminaries

Introduction

Some Examples

Brownian Motions and Martingales

Stochastic Integrals

Stochastic Differential Equations of Itô Type

Lévy Processes and Stochastic Integrals

Stochastic Differential Equations of Lévy Type

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 Stochastic Heat Equation

Linear Equations with Additive Noise

Some Regularity Properties

Stochastic Reaction–Diffusion Equations

Parabolic Equations with Gradient-Dependent Noise

Nonlinear Parabolic Equations with Lévy-Type Noise

Stochastic Parabolic Equations in the Whole Space

Introduction

Preliminaries

Linear and Semilinear 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 an 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 the 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

Spatial Population Growth Model in Random Environment

HJMM Equation in Finance

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

Bibliography

Index

Subject Categories

BISAC Subject Codes/Headings:
MAT007000
MATHEMATICS / Differential Equations
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis
SCI040000
SCIENCE / Mathematical Physics