Laplace approximations for large deviations of diffusion processes on Euclidean spaces

Abstract

Consider a class of uniformly elliptic diffusion processes $\{X_t{\}}_{t\ge 0}$ on Euclidean spaces ${\mathbf{R}}^d$. We give an estimate of $E^{P_x}\left[\exp \left(T\mathrm{\Phi }\right(1/T{\int }_0^T{\delta }_{X_t}dt\left)\right)|X_T=y\right]$ as $T\to \mathrm{\infty }$ up to the order $1+o\left(1\right)$, where ${\delta }_{\cdot }$ means the delta measure, and $\mathrm{\Phi }$ is a function on the set of measures on ${\mathbf{R}}^d$. This is a generalization of the works by Bolthausen-Deuschel-Tamura [3] and Kusuoka-Liang [10], which studied the same problems for processes on compact state spaces.