http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
Kolmogorov Equations for Stochastic PDEs
The subject of this book is stochastic partial differential equations, in particular, reaction-diffusion equations, Burgers and Navier-Stokes equations and the corresponding Kolmogorov equations. For each case the transition semigroup is considered and irreducibility, the strong Feller property, and invariant measures are investigated. Moreover, it is proved that the exponential functions provide a core for the infinitesimal generator. As a consequence, it is possible to study Sobolev spaces with respect to invariant measures and to prove a basic formula of integration by parts (the so-called "carr du champs identity". Several results were proved by the author and his collaborators and appear in book form for the first time. Presenting the basic elements of the theory in a simple and compact way, the book covers a one-year course directed to graduate students in mathematics or physics. The only prerequisites are basic probability (including finite dimensional stochastic differential equations), basic functional analysis and some elements of the theory of partial differential equations.
Introduction and Preliminaries -- Stochastic Perturbations of Linear Equations -- Stochastic Differential Equations with Lipschitz Nonlinearities -- Reaction-Diffusion Equations -- The Stochastic Burgers Equation -- The Stochastic 2D Navier-Stokes Equation -- Bibliography -- Index.
Advanced Courses in Mathematics CRM Barcelona, Centre de Recerca Matemàtica