Stochastic Calculus of Variation for Wiener Functionals

Abstract

The most important Gaussian probability space in applications is the classical Wiener space. The most important Wiener functionals are so-called Itô functionals, namely the Itô integrals and solutions of Itô stochastic differential equations. In this chapter, we make an exposition for the theory of stochastic calculus of variation for Wiener functionals and its applications to regularities of fundamental solutions for parabolic partial differential equations, especially to the probabilistic proof of Hörmander’s theorem on hypoellipticity of partial differential operators. Moreover, we introduce two important branches in this area which were developed very recently: the quasi sure analysis and the anticipating stochastic calculus.