Large Deviations for the Langevin equation with strong damping

TL;DR

This paper investigates large deviations in Langevin dynamics with strong, state-dependent damping and noise, using weak convergence methods to derive the action functional and applying results to exit problems and wave front propagation.

Contribution

It introduces a novel analysis of large deviations for Langevin equations with strong, state-dependent damping using weak convergence techniques.

Findings

Derived the large deviations principle for Langevin equations with strong damping

Identified the explicit form of the action functional in this regime

Applied results to exit problems and wave front propagation scenarios

Abstract

We study large deviations in the Langevin dynamics, with damping of order $\e^{-1}$ and noise of order $1$, as $\e\downarrow 0$. The damping coefficient is assumed to be state dependent. We proceed first with a change of time and then, we use a weak convergence approach to large deviations and their equivalent formulation in terms of the Laplace principle, to determine the good action functional. Some applications of these results to the exit problem from a domain and to the wave front propagation for a suitable class of reaction diffusion equations are considered.