Laplace approximations for large deviations of diffusion processes on Euclidean spaces

Abstract

Consider a class of uniformly elliptic diffusion processes { X_t }_{t ≥ 0} on Euclidean spaces \bm{R}^d . We give an estimate of E^P_x \bigl[ exp (T ¶hi ({1}/{T} ∈t_0^T δ_{X_t} dt)) \big| X_T =y \bigr] as T → ∞ up to the order 1 + o(1), where δ_• means the delta measure, and ¶hi is a function on the set of measures on \bm{R}^d . This is a generalization of the works by Bolthausen-Deuschel-Tamura \cite{B-D-T} and Kusuoka-Liang \cite{K-L_Torus}, which studied the same problems for processes on compact state spaces.