Taylor expansion of the density in a stochastic heat equation

Abstract

We prove a general result on asymptotic expansions of densities for families of perturbed Wiener functionals. As an application, we consider a stochastic heat equation driven by a space-time white noise εW˙ t,x, ε ∈ (0, 1]. The main theorem describes the asymptotics, as ε ↓ 0, of the density pε t,x(y) of the solution at a fixed point (t, x) for some particular value y ∈ R, which, in the diffusion case, plays the role of the diagonal.