Laplace Approximations for Diffusion Processes on Torus: Nondegenerate Case

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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 \nabla $, 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.
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