Laplace Approximations for Diffusion Processes on Torus: Nondegenerate Case
Vol. 8 (2001), No. 1, Page 43--70.
Kusuoka, Shigeo ; Liang, Song
Laplace Approximations for Diffusion Processes on Torus: Nondegenerate Case
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Abstract:
Let Td=Rd/Zd, and consider the family of probability measures {Px}x∈Td on C([0,∞);Td) 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 Φ be a mapping {\cal M} ({\bf T}^d) \to {\bf R} . Under a nuclearity assumption on the second Fréchet 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.
Mathematics Subject Classification (2000): Primary 60F10; Secondary 60J60
Mathematical Reviews Number: MR1818905
Received: 2000-01-24
