Stochastic Analysis

This book accounts in 5 independent parts, recent main developments of Stochastic Analysis: Gross-Stroock Sobolev space over a Gaussian probability space; quasi-sure analysis; anticipate stochastic integrals as divergence operators; principle of transfer from ordinary differential equations to stochastic differential equations; Malliavin calculus and elliptic estimates; stochastic Analysis in infinite dimension. Abstract: This book accounts in 5 independent parts, recent main developments of Stochastic Analysis: Gross-Stroock Sobolev space over a Gaussian probability space; quasi-sure analysis; anticipate stochastic integrals as divergence operators; principle of transfer from ordinary differential equations to stochastic differential equations; Malliavin calculus and elliptic estimates; stochastic Analysis in infinite dimension Front Matter....Pages I-XI Front Matter....Pages 1-2 Gaussian Probability Spaces....Pages 3-27 Gross-Stroock Sobolev Spaces over a Gaussian Probability Space....Pages 29-62 Smoothness of Laws....Pages 63-83 Front Matter....Pages 87-88 Foundations of Quasi-Sure Analysis: Hierarchy of Capacities and Precise Gaussian Probability Spaces....Pages 89-123 Differential Geometry on a Precise Gaussian Probability Space....Pages 125-144 Front Matter....Pages 147-148 White Noise Stochastic Integrals as Divergences....Pages 149-171 Itô’s Theory of Stochastic Integration....Pages 173-198 Front Matter....Pages 201-202 From Ordinary Differential Equations to Stochastic Flow: The Transfer Principle....Pages 203-235 Elliptic Estimates Through Stochastic Analysis....Pages 237-251 Front Matter....Pages 257-258 Stochastic Analysis on Wiener Spaces....Pages 259-271 Path Spaces and Their Tangent Spaces....Pages 273-298 Back Matter....Pages 301-346