2nd Edition

Stochastic Partial Differential Equations

By Pao-Liu Chow Copyright 2015
    334 Pages
    by Chapman & Hall

    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.

    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

    Biography

    Pao-Liu Chow

    "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