Asymptotic behaviour of the density in a parabolic SPDE

Abstract

Consider the density of the solution $X(t, x)$ of a stochastic heat equation with small noise at a fixed $t \in[0, T], x \in[0,1]$. In this paper we study the asymptotics of this density as the noise vanishes. A kind of Taylor expansion in powers of the noise parameter is obtained. The coefficients and the residue of the expansion are explicitly calculated. In order to obtain this result some type of exponential estimates of tail probabilities of the difference between the approximating process and the limit one is proved. Also a suitable iterative local integration by parts formula is developed.