A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures

TL;DR

This paper links Freidlin-Wentzell large deviation theory with weak KAM solutions, explicitly characterizing key concepts and showing how the rate function corresponds to a unique viscosity solution of the stationary Hamilton-Jacobi equation, serving as the energy landscape.

Contribution

It provides a novel weak KAM perspective on Freidlin-Wentzell's variational construction of the rate function for invariant measures, explicitly characterizing essential weak KAM concepts.

Findings

Explicit characterization of Peierls barrier and Aubry/Mather sets in a 1D diffusion on a torus

Connection between the rate function and the maximal Lipschitz viscosity solution of HJE

A new method for selecting a unique weak KAM solution via boundary data and vanishing viscosity

Abstract

This paper reinterprets Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. Through a one-dimensional irreversible diffusion process on a torus, we explicitly characterize essential concepts in the weak KAM theory, such as the Peierls barrier and the projected Mather/Aubry/Ma\~n\'e sets. The weak KAM representation of Freidlin-Wentzell's variational construction of the rate function is discussed based on the global adjustment for the boundary data on the Aubry set and the local trimming from the lifted Peierls barriers. This rate function gives the maximal Lipschitz continuous viscosity solution to the corresponding stationary Hamilton-Jacobi equation (HJE), satisfying Freidlin-Wentzell's variational formula for the boundary data. Choosing meaningful self-consistent boundary data at each local attractor are essential to select a unique weak KAM solution to stationary HJE. This selected viscosity solution also serves as the global energy landscape of the original stochastic process. This selection for stationary HJEs can be described by first taking the long time limit and then taking the zero noise limit, which also provides a special construction of vanishing viscosity approximation.