As a relatively new area in mathematics, stochastic partial differential equations (PDEs) are still at a tender age and have not yet received much attention in the mathematical community. Filling the void of an introductory text in the field, Stochastic Partial Differential Equations introduces PDEs to students familiar with basic probability theory and Ito's equations, highlighting several computational and analytical techniques. Without assuming specific knowledge of PDEs, the text includes many challenging problems in stochastic analysis and treats stochastic PDEs in a practical way. The author first brings the subject back to its root in classical concrete problems. He then discusses a unified theory of stochastic evolution equations and describes a few applied problems, including the random vibration of a nonlinear elastic beam and invariant measures for stochastic Navier-Stokes equations. The book concludes by pointing out the connection of stochastic PDEs to infinite-dimensional stochastic analysis.
By thoroughly covering the concepts and applications of stochastic PDEs at an introductory level, this text provides a guide to current research topics and lays the groundwork for further study.
PREFACE PRELIMINARIES Introduction Some Examples Brownian Motions and Martingales Stochastic Integrals Stochastic Differential Equations Comments SCALAR EQUATIONS OF FIRST ORDER Introduction Generalized Ito's Formula Linear Stochastic Equations Quasilinear Equations General Remarks STOCHASTIC PARABOLIC EQUATIONS Introduction Preliminaries Solution of Random Heat Equation Linear Equations with Additive Noise Some Regularity Properties Random Reaction-Diffusion Equations Parabolic Equations with Gradient-Dependent Noise STOCHASTIC PARABOLIC EQUATIONS IN THE WHOLE SPACE Introduction Preliminaries Linear and Similinear 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 Unbounded Domain Randomly Perturbed Hyperbolic Systems STOCHASTIC EVOLUTION EQUATIONS IN HILBERT SPACES Introduction Hilbert Space-Valued Martingales Stochastic Integrals in Hilbert Spaces Ito's Formula Stochastic Evolution Equations Mild Solutions Strong Solutions Stochastic Evolution Equations of Second Order ASYMPTOTIC BEHAVIOR OF SOLUTIONS Introduction Ito'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 Schrodinger Equation Nonlinear Stochastic Beam Equations Stochastic Stability of Cahn-Hilliard Equation Invariant Measures for Stochastic Navier-Stokes Equations 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 REFERENCES INDEX