The Large Deviation Principle for Stochastic Flow of Stochastic Slow-Fast Motions

TL;DR

This paper establishes a large deviation principle for a fully coupled stochastic slow-fast system with non-Lipschitz slow dynamics, using advanced probabilistic techniques.

Contribution

It proves the existence of the stochastic flow and the large deviation principle for the slow variable under non-Lipschitz conditions, extending previous results.

Findings

Existence of stochastic flow for the slow variable.

Large deviation principle established for the slow variable.

Methodology based on averaging, variational representation, and weak convergence.

Abstract

In this paper, we consider a kind of fully coupled slow fast motion, in which the slow variable satisfies the non Lipschitz condition. We prove that the stochastic flow of the slow variable exists and moreover, satisfies the large deviation principle. The argument is mainly based on Khasminskii's averaging principle, the variational representation of the exponential functional of the Brownian motion, and the weak convergence framework proposed by Budhiraja and Dupuis.