Stochastic PDEs

David Nualart's Abstract
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Title: An introduction to Malliavin Calculus and its applications to SPDEs

Abstract: The aim of these lectures is to first introduce the stochastic calculus of variations (Malliavin Calculus) for a Gaussian process and to establish the basic criteria for the existence and smoothness of densities. We will present two types of applications of Malliavin Calculus to SPDEs:
  1. Absolute continuity and smoothness of the density for the law of the solution at a fixed time of the stochastic heat and wave equations, assuming a nondegeneracy condition on the diffusion coefficient.
  2. Existence and uniqueness of a solution for the one-dimensional stochastic heat equation driven by a space-time white noise, when the diffusion coefficient is bounded away from zero and the drift is irregular (regularization effect).

Prerequisites: Good working knowledge of Ito stochastic calculus and basic properties of Gaussian processes. Some basic properties of the stochastic heat and wave equations.
VIGRE       U of U Math