Expansion of the density: a Wiener-chaos approach

Abstract

We prove a Taylor expansion of the density pε(y) of a Wiener functional Fε with Wiener-chaos decomposition Fε=y+∑∞n=1εnIn(fn), ε∈(0,1]. Using Malliavin calculus, a precise description of the coefficients in the development in terms of the multiple integrals In(fn) is provided. This general result is applied to the study of the density in two examples of hyperbolic stochastic partial differential equations with linear coefficients, where the driving noise has been perturbed by a coefficient ε.