Large deviations for rough path lifts of Watanabe's pullbacks of delta functions

TL;DR

This paper establishes a large deviation principle for rough path lifts of Watanabe's delta functions associated with hypoelliptic diffusions, generalizing Freidlin-Wentzell theory to a rough path setting.

Contribution

It proves a Schilder-type large deviation principle for the rough path lifts of Watanabe's delta functions, extending large deviation results to hypoelliptic diffusion processes.

Findings

Large deviation principle for lifted measures as scale tends to zero

Corollary confirming a conjecture by Takanobu and Watanabe

Generalization of Freidlin-Wentzell large deviations for pinned diffusions

Abstract

We study Donsker-Watanabe's delta functions associated with strongly hypoelliptic diffusion processes indexed by a small parameter. They are finite Borel measures on the Wiener space and admit a rough path lift. Our main result is a large deviation principle of Schilder type for the lifted measures on the geometric rough path space as the scale parameter tends to zero. As a corollary, we obtain a large deviation principle conjectured by Takanobu and Watanabe, which is a generalization of a large deviation principle of Freidlin-Wentzell type for pinned diffusion processes.