Exponential ergodicity of L\'{e}vy driven Langevin dynamics with singular potentials

TL;DR

This paper proves exponential ergodicity for Lévy-driven Langevin dynamics with singular potentials, including Lennard-Jones and Coulomb interactions, using novel Lyapunov functions and advanced stochastic analysis techniques.

Contribution

It introduces new methods for establishing ergodicity in Lévy-driven Langevin systems with singular potentials, extending previous results beyond Brownian-driven cases.

Findings

Proves exponential ergodicity for systems with Lennard-Jones potentials.

Extends ergodicity results to Coulomb potentials.

Develops novel Lyapunov functions for non-local operators.

Abstract

In this paper, we address exponential ergodicity for L\'{e}vy driven Langevin dynamics with singular potentials, which can be used to model the time evolution of a molecular system consisting of $N$ particles moving in $\R^d$ and subject to discontinuous stochastic forces. In particular, our results are applicable to the singular setups concerned with not only the Lennard-Jones-like interaction potentials but also the Coulomb potentials. In addition to Harris' theorem, the approach is based on novel constructions of proper Lyapunov functions (which are completely different from the setting for Langevin dynamics driven by Brownian motions), on invoking the H\"{o}rmander theorem for non-local operators and on solving the issue on an approximate controllability of the associated deterministic system as well as on exploiting the time-change idea.