Short time asymptotics of random heat kernels

We study the random fundamental solution of a linear stochastic parabolic partial differential equation of the second order. We are particularly interested in the short time asymptotics of the random fundamental solution. If the random differential operator is nondegenerate, it will be shown that the short time asymptotics is represented by a Gaussian kernel. However, if it is degenerate, the short time asymptotics will be represented by a joint distribution of a certain linear sum of a Brownian motion and its multiple Wiener-Stratonovich integrals. Our approach is based on the asymptotic analysis of the stochastic flow which solves the stochastic partial differential equation. It is parallel to the study of the short time asymptotics of the fundamental solution of a second order (deterministic) partial differerntial equation in Kunita [3].