Laplace approximations for large deviations of diffusion processes on Euclidean spaces

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(TΦ(1/T ∫^T₀ δ X_t dt))||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.