Since T. Lyons invented rough path theory, one of its most successful applications is a new proof of Freidlin-Wentzell's large deviation principle for diffusion processes. In this paper we extend this method to the case of pinned diffusion processes under a mild ellipticity assumption. Besides rough path theory, our main tool is quasi-sure analysis, which is a kind of potential theory in Malliavin calculus.

In this paper we establish a Taylor-like expansion in the context of the rough path theory for a family of Itô maps indexed by a small parameter. We treat not only the case that the roughness p satisfies [p] = 2, but also the case that [p] ≥ 3. As an application, we discuss the Laplace asymptotics for Itô functionals of Brownian rough paths.This rough path is called be the smooth rough path lying above X ∈ BV(V) and is again denoted by X (when there is no possibility of confusion). The d-closure of the totality of all the smooth rough paths is denoted by GΩ p (V), which is called the space of geometric rough paths. This is a complete metric space. (If V is separable, then GΩ p (V) is also separable, which can easily be seen from Corollary 2.3 below.)2.2 On basic properties of q-variational paths (1 ≤ q < 2).Let 1 ≤ q < 2. For a real Banach space V, set C 0,q (V) = {X ∈ C([0, 1], V) | X 0 = 0 and X q < ∞},

In this paper, we establish asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths under the condition that the phase function has finitely many non-degenerate minima. Our main tool is the Banach space-valued rough path theory of T. Lyons. We use a large deviation principle and the stochastic Taylor expansion with respect to the topology of the space of geometric rough paths. This is a continuation of a series of papers by Inahama [Y. Inahama, Laplace's method for the laws of heat processes on loop spaces, J. Funct. Anal. 232 (2006) 148-194] and by Inahama and Kawabi [Y. Inahama, H. Kawabi, Large deviations for heat kernel measures on loop spaces via rough paths, J. London Math. Soc. 73 (3) (2006) 797-816], [Y. Inahama, H. Kawabi, On asymptotics of certain Banach spacevalued Itô functionals of Brownian rough paths, in:

The main objective of this paper is to establish a large deviation principle for heat kernel measures on loop spaces. It gives an extension of Fang and Zhang's results on loop groups. For the proof, we use the continuity theorem of Lyons' rough path theory.