On the small time asymptotics of diffusion processes on Hilbert spaces

Abstract

In this paper,we establish a small time large deviation principle and obtain the following small time asymptotics:

\lim_{t \to 0}2t \log P(X_0 \in B, X_t \in C) = -d^2 (B, C),

for diffusion processes on Hilbert spaces, where $d(B,C)$ is the intrinsic metric between two subsets $B$ and $C$ associated with the diffusions. The case of perturbed Ornstein–Uhlenbeck processes is treated separately at the end of the paper.