Large deviation principle of Freidlin-Wentzell type for pinned diffusion processes

Abstract

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.

Large Deviations for Rough Path Lifts of Watanabe's Pullbacks of Delta Functions

“…This work is a generalization of it to the strongly hypoelliptic case. Note that many basic results on quasi-sure analysis for Brownian rough path were already obtained in [15]. Compared to [15], the lower estimate becomes more difficult, while the upper estimate remains somewhat similar.…”

“…The elliptic case was already done in the author's previous work [15]. This work is a generalization of it to the strongly hypoelliptic case.…”

“…This class includes fractional Brownian rough path with Hurst parameter H ∈ (1/4, 1/2]. The original motivation of [15] was to extend the idea in [19] to the case of pinned diffusion measures.) Another advantage of this method is that one can also prove Laplace approximation (i.e., the precise asymptotics of LDP of Freidlin-Wentzell type) along the same streamline with or without Malliavin calculus.…”

“…This is a key lemma in the proof of the lower estimate of our main theorem. In Section 4 we present some preliminaries on quasi-sure analysis on rough path space, all of which were already shown or used in [15].…”

“…Compared to the elliptic case in [15], this part becomes more difficult for two reasons. (These are closely related, however.)…”