Large-deviation theory for a Brownian particle on a ring: a WKB approach

TL;DR

This paper develops a WKB method to analyze how the large deviation function of a Brownian particle on a ring transitions from a cusp at zero velocity in the zero noise limit to a smooth curve in the weak noise limit, enhancing understanding of stochastic fluctuations.

Contribution

It introduces a WKB approach to study the smoothing of the large deviation function's cusp for a Brownian particle on a ring in the weak noise regime.

Findings

The cusp in the large deviation function is rounded in the weak noise limit.

The WKB method provides a detailed analysis of the transition from cusp to smoothness.

The approach offers insights into the stochastic behavior of confined Brownian particles.

Abstract

We study the large deviation function of the displacement of a Brownian particle confined on a ring. In the zero noise limit this large deviation function has a cusp at zero velocity given by the Freidlin-Wentzell theory. We develop a WKB approach to analyse how this cusp is rounded in the weak noise limit.