Stochastic Partial Differential Equations

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