Stochastic partial differential equations

Table of Contents

1. Preliminaries

i. Introduction
ii. Some Examples
iii. Brownian Motions and Martingales
iv. Stochastic Integrals
v. Stochastic Differential Equations of Itô Type
vi. Lévy Processes and Stochastic Integrals
vii. Stochastic Differential Equations of Lévy Type
viii. Comments

2. Scalar Equations of First Order

i. Introduction
ii. Generalized Itô’s Formula
iii. Linear Stochastic Equations
iv. Quasilinear Equations
v. General Remarks

3. Stochastic Parabolic Equations

i. Introduction
ii. Preliminaries
iii. Solution of Stochastic Heat Equation
iv. Linear Equations with Additive Noise
v. Some Regularity Properties
vi. Stochastic Reaction–Diffusion Equations
vii. Parabolic Equations with Gradient-Dependent Noise
viii. Nonlinear Parabolic Equations with Lévy-Type Noise

4. Stochastic Parabolic Equations in the Whole Space

i. Introduction
ii. Preliminaries
iii. Linear and Semilinear Equations
iv. Feynman–Kac Formula
v. Positivity of Solutions
vi. Correlation Functions of Solutions

5. Stochastic Hyperbolic Equations

i. Introduction
ii. Preliminaries
iii. Wave Equation with Additive Noise
iv. Semilinear Wave Equations
v. Wave Equations in an Unbounded Domain
vi. Randomly Perturbed Hyperbolic Systems

6. Stochastic Evolution Equations in Hilbert Spaces

i. Introduction
ii. Hilbert Space–Valued Martingales
iii. Stochastic Integrals in Hilbert Spaces
iv. Itô’s Formula
v. Stochastic Evolution Equations
vi. Mild Solutions
vii. Strong Solutions
viii. Stochastic Evolution Equations of the Second Order

7. Asymptotic Behavior of Solutions

i. Introduction
ii. Itô’s Formula and Lyapunov Functionals
iii. Boundedness of Solutions
iv. Stability of Null Solution
v. Invariant Measures
vi. Small Random Perturbation Problems
vii. Large Deviations Problems

8. Further Applications

i. Introduction
ii. Stochastic Burgers and Related Equations
iii. Random Schrödinger Equation
iv. Nonlinear Stochastic Beam Equations
v. Stochastic Stability of Cahn–Hilliard Equation
vi. Invariant Measures for Stochastic Navier–Stokes Equations
vii. Spatial Population Growth Model in Random Environment
viii. HJMM Equation in Finance

9. Diffusion Equations in Infinite Dimensions

i. Introduction
ii. Diffusion Processes and Kolmogorov Equations
iii. Gauss–Sobolev Spaces
iv. Ornstein–Uhlenbeck Semigroup
v. Parabolic Equations and Related Elliptic Problems
vi. Characteristic Functionals and Hopf Equations

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.


(https://www.crcpress.com/Stochastic-Partial-Differential-Equations-Second-Edition/Chow/9781466579552)