Hypoelliptic multiscale Langevin diffusions: Large deviations, invariant measures and small mass asymptotics

TL;DR

This paper investigates large deviations and small mass asymptotics for hypoelliptic Langevin diffusions, providing new asymptotic expansions and convergence results that extend understanding beyond gradient-driven systems.

Contribution

It introduces a comprehensive analysis of large deviations and invariant measure asymptotics for non-gradient hypoelliptic Langevin diffusions, including explicit bounds and convergence rates.

Findings

Large deviations behavior aligns with overdamped systems in the small mass limit.

Asymptotic expansion of invariant measure density with respect to mass parameter.

Improved hypocoercivity results for kinetic Fokker-Planck equations.

Abstract

We consider a general class of non-gradient hypoelliptic Langevin diffusions and study two related questions. The first one is large deviations for hypoelliptic multiscale diffusions. The second one is small mass asymptotics of the invariant measure corresponding to hypoelliptic Langevin operators and of related hypoelliptic Poisson equations. The invariant measure corresponding to the hypoelliptic problem and appropriate hypoelliptic Poisson equations enter the large deviations rate function due to the multiscale effects. Based on the small mass asymptotics we derive that the large deviations behavior of the multiscale hypoelliptic diffusion is consistent with the large deviations behavior of its overdamped counterpart. Additionally, we rigorously obtain an asymptotic expansion of the solution to the related density of the invariant measure and to hypoelliptic Poisson equations with respect to the mass parameter, characterizing the order of convergence. The proof of convergence of invariant measures is of independent interest, as it involves an improvement of the hypocoercivity result for the kinetic Fokker-Planck equation. We do not restrict attention to gradient drifts and our proof provides explicit information on the dependence of the bounds of interest in terms of the mass parameter.