Malliavin calculus is a stochastic calculus of variations on the Wiener space. The main results of this theory are currently having influence on research developments at the cross section of probability and infinite-dimensional analysis. On the applied level, Malliavin calculus is used, for example, in the study by probabilistic methods of mathematical models in finance. This book presents some applications of Malliavin calculus to stochastic partial differential equations driven by Gaussian noises. The first five chapters are devoted to an introduction of the calculus itself, based on a general Gaussian space. In the last chapters of the book, recent research on regularity of the solution of stochastic partial differential equations, and the existence and smoothness of their probability laws, are discussed.
Malliavin calculus is a stochastic calculus of variations of the Wiener space. On the applied level, Malliavin calculus is used, for example, in probabilistic numerical methods in financial mathematics. The aim of this book is to provide applications of Malliavin calculus to the probability laws of solutions of stochastic partial differential equations driven by Gaussian noises. The first five chapters are devoted to the introduction of the calculus itself based on a general Gaussian space. The last chapters of the book are devoted to the applications to stochastic partial differential equations based on recent research.
Integration by Parts and Absolute Continuity of Probability Laws - Finite Dimensional Malliavin Calculus - The Basic Operators of Malliavin Calculus - Representation of Wiener Functionals - riteria for Absolute Continuity - Stochastic Partial Differnetial Equations Driven by Spatially Homogeneous Gaussian Noise - Malliavin Regularity of Solutions of SPDE's - Analysis of the Malliavin Matrix of Solutions of the SPDE's - Definitions of spaces - Bibliography.
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