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2nd Edition

Stochastic Partial Differential Equations




ISBN 9781466579552
Published December 10, 2014 by Chapman and Hall/CRC
334 Pages

 
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Book 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.

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

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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