A representation formula for large deviations rate functionals of invariant measures on the one dimensional torus

TL;DR

This paper derives a simple, geometric representation formula for the large deviations rate functional of invariant measures for 1D diffusions and Markov processes, simplifying complex previous characterizations and revealing a universal transformation.

Contribution

It introduces a geometric transformation-based formula for the large deviations rate functional, simplifying prior complex optimization characterizations and establishing universality for Hamilton-Jacobi solutions.

Findings

Derived a simple geometric formula for the rate functional.

Extended the formula to piecewise deterministic Markov processes.

Proved universality of the transformation for Hamilton-Jacobi equations.

Abstract

We consider a generic diffusion on the 1D torus and give a simple representation formula for the large deviation rate functional of its invariant probability measure, in the limit of vanishing noise. Previously, this rate functional had been characterized by M.I. Freidlin and A.D.\ Wentzell as solution of a rather complex optimization problem. We discuss this last problem in full generality and show that it leads to our formula. We express the rate functional by means of a geometric transformation that, with a Maxwell-like construction, creates flat regions. We then consider piecewise deterministic Markov processes on the 1D torus and show that the corresponding large deviation rate functional for the stationary distribution is obtained by applying the same transformation. Inspired by this, we prove a universality result showing that the transformation generates viscosity solution of stationary Hamilton--Jacobi equation associated to any Hamiltonian $H$ satisfying suitable weak conditions.