## Laplace Approximations for Diffusion Processes on Torus: Nondegenerate Case

J. Math. Sci. Univ. Tokyo
Vol. 8 (2001), No. 1, Page 43--70.

Kusuoka, Shigeo ; Liang, Song
Laplace Approximations for Diffusion Processes on Torus: Nondegenerate Case
Let ${\bf T}^d = {\bf R}^d / {\bf Z}^d$, and consider the family of probability measures $\{ P_x \}_{x \in {\bf T}^d}$ on $C([0, \infty); {\bf T}^d)$ given by the infinitesimal generator $L_0 \equiv \frac{1}{2} Î + b \cdot \nabla$, where $b: {\bf T}^d \to {\bf R}^d$ is a continuous function. Let $Î¦$ be a mapping ${\cal M} ({\bf T}^d) \to {\bf R}$. Under a nuclearity assumption on the second FrÃ©chet differential of $Î¦$, an asymptotic evaluation of $Z_T^{x, y} \equiv E^{P_x} \left[ \exp \left( T Î¦ (\frac{1}{T} \int_0^T Î´_{X_t} dt)\right) \Big| X_T = y \right]$, up to a factor $(1 + o(1))$, has been gotten in Bolthausen-Deuschel-Tamura \cite{B-D-T}. In this paper, we show that the same asymptotic evaluation holds without the nuclearity assumption.