Second-Order Fast-Slow Stochastic Systems

TL;DR

This paper develops large deviations principles for second-order stochastic systems with multiple scales, relaxing traditional assumptions and applying to models in physics and statistical mechanics.

Contribution

It introduces a novel asymptotic analysis framework for second-order stochastic systems under weaker conditions, including non-Lipschitz diffusion coefficients.

Findings

Established large deviations principles for systems with diffusion fast processes without Lipschitz assumptions.

Extended results to general setups with no specific structure, relying on local large deviations of first-order systems.

Provided theoretical foundations applicable to Langevin dynamics and stochastic acceleration models.

Abstract

This paper focuses on systems of nonlinear second-order stochastic differential equations with multi-scales. The motivation for our study stems from mathematical physics and statistical mechanics, for examples, Langevin dynamics and stochastic acceleration in a random environment. Our effort is to carry out asymptotic analysis to establish large deviations principles. Our focus is on obtaining the desired results for systems under weaker conditions. When the fast-varying process is a diffusion, neither Lipschitz continuity nor linear growth needs to be assumed. Our approach is based on combinations of the intuition from Smoluchowski-Kramers approximation, and the methods initiated in [34] relying on the concepts of relatively large deviations compactness and the identification of rate functions. When the fast-varying process is under a general setup with no specified structure, the paper establishes the large deviations principle of the underlying system under the assumption on the local large deviations principles of the corresponding first-order system.