Laplace approximations for large deviations of diffusion processes on Euclidean spaces

Abstract

Consider a class of uniformly elliptic diffusion processes { X t } t ≥ 0 on Euclidean spaces R d . We give an estimate of E P x exp ( T Φ ( 1 / T ∫ 0 T δ X t d t ) ) | X T = y as T → ∞ up to the order 1 + o ( 1 ) , where δ ⋅ means the delta measure, and Φ is a function on the set of measures on 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.