Large Deviations for Small Noise Diffusions Over Long Time

TL;DR

This paper investigates large deviation principles for small noise diffusion processes over long times, focusing on empirical measures and slow-fast systems with degeneracy, providing explicit rate functions and new analytical methods.

Contribution

It develops new large deviation results for degenerate diffusion processes over long times, extending classical theories to cases with vanishing noise and degeneracy.

Findings

Established large deviation principles for empirical measures with degenerate noise.

Derived explicit rate functions with simple forms for the studied systems.

Provided a degenerate averaging principle for slow-fast diffusions with vanishing noise.

Abstract

We study two problems. First, we consider the large deviation behavior of empirical measures of certain diffusion processes as, simultaneously, the time horizon becomes large and noise becomes vanishingly small. The law of large numbers (LLN) of the empirical measure in this asymptotic regime is given by the unique equilibrium of the noiseless dynamics. Due to degeneracy of the noise in the limit, the methods of Donsker and Varadhan (1976) are not directly applicable and new ideas are needed. Second, we study a system of slow-fast diffusions where both the slow and the fast components have vanishing noise on their natural time scales. This time the LLN is governed by a degenerate averaging principle in which local equilibria of the noiseless system obtained from the fast dynamics describe the asymptotic evolution of the slow component. We establish a large deviation principle that describes probabilities of divergence from this behavior. On the one hand our methods require stronger assumptions than the nondegenerate settings, while on the other hand the rate functions take simple and explicit forms that have striking differences from their nondegenerate counterparts.