Expansion of the density: a Wiener-chaos approach

Abstract

We prove a Taylor expansion of the density $p^{\epsilon }\left(y\right)$ of a Wiener functional $F^{\epsilon }$ with Wiener-chaos decomposition $F^{\epsilon }=y+{\sum }_{n=1}^{\mathrm{\infty }}{\epsilon }_n^{\mathrm{nI}}\left(f_n\right)$, $\epsilon \in (0,1]$. Using Malliavin calculus, a precise description of the coefficients in the development in terms of the multiple integrals $I_n\left(f_n\right)$ 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 $\epsilon$.