Asymptotic Behaviour of the Density in a Parabolic SPDE

Abstract

Consider the density of the solution Xt; x of a stochastic heat equation with small noise at a xed t 2 0; T , x 2 0; 1 . In the paper we study the asymptotics of this density as the noise is vanishing. A kind of Taylor expansion in powers of the noise parameter is obtained. The coe cients 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 di erence between the approximating process and the limit one is proved. Also a suitable local integration by parts formula is developped.

Lower bounds for densities of uniformly elliptic random variables on Wiener space

“…The Varadhan estimates for the stochastic heat equation were obtained in [9] using a general theorem taken from [16]. It is clear that these two results are deeply related.…”

Density estimates on a parabolic SPDE

“…g = 0) have been found by Millet and Sanz-Solé [19]. In this case, we refer to [13] for more information about the set { h …”