Analysis of Wiener Functionals (Malliavin Calculus) and its Applications to Heat Kernels

An analysis of Wiener functionals is studied as a kind of Schwartz distribution theory on Wiener space. For this, we introduce, besides ordinary $L_p$-spaces of Wiener functionals, Sobolev-type spaces of (generalized) Wiener functionals. Any Schwartz distribution on $\mathbf{R}^d$ is pulled back to a generalized Wiener functional by a $d$-dimensional Wiener map which is smooth and nondegenerate in the sense of Malliavin. As applications, we construct a heat kernel (i.e., the fundamental solution of a heat equation) by a generalized expectation of the Dirac delta function pulled back by an Ito map, i.e., a Wiener map obtained by solving Ito's stochastic differential equations. Short-time asymptotics of heat kernels are studied through the asymptotics, in terms of Sobolev norms, of the generalized Wiener functional under the expectation.