Laplace Approximations for Diffusion Processes on Torus: Nondegenerate Case
Shigeo Kusuoka, Liang Song · Institutional Repositories DataBase (IRDB) · 2001
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} Δ + b \cdot \ abla $, where $b: {\bf T}^d \to {\bf R}^d $ is a continuous function. Let $ Φ $ be a mapping $ {\cal M} ({\bf T}^d) \to {\bf R} $. Under a nuclearity assumption on the second Frechet differential of $ Φ $, an asymptotic evaluation of $ Z_T^{x, y} \equiv E^{P_x} \left[ \exp \left( T Φ (\frac{1}{T} \int_0^T δ_{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.