Developed in the 1970s to study the existence and smoothness of density for the probability laws of random vectors, Malliavin calculus--a stochastic calculus of variation on the Wiener space--has proven fruitful in many problems in probability theory, particularly in probabilistic numerical methods in financial mathematics. This book present

Presented in a comprehensive way, Malliavin Calculus with Applications to Stochastic Partial Differential Equations describes applications of Malliavin calculus to the analysis of probability laws of solutions of stochastic partial differential equations, driven by Gaussian noises that are white in time and colored in space. The text begins with an introduction to this type of calculus based on a general Gaussian space, from finite-dimensional to infinite-dimensional settings. The book later presents applications to stochastic partial differential equations based on current research, supplemented by comments concerning the origin of the work developed within and its references.

Introduction. Integration by Parts and Absolute Continuity of Probability Laws. Finite Dimensional Malliavin Calculus. The Basic Operators of Malliavin Calculus. Representation of Wiener Functionals. Criteria for Absolute Continuity and Smoothness of Probability Laws. Stochastic Partial Differential Equations driven by Spatially Homogenous Gaussian Noise. Malliavin Regularity of Solutions of SPDEs. Analysis of the Malliavin Matrix of Solutions of SPDEs. Definition of Spaces Used Throughout the Course.