L2 geometric ergodicity for the kinetic Langevin process with non-equilibrium steady states

TL;DR

This paper establishes L2 convergence rates for the kinetic Langevin process in non-equilibrium steady states where the invariant measure lacks an explicit density, extending previous results limited to equilibrium cases.

Contribution

It provides the first L2 geometric ergodicity results for kinetic Langevin processes with non-equilibrium steady states, broadening the understanding beyond conservative force scenarios.

Findings

Proves L2 convergence rates for non-equilibrium steady states

Extends geometric ergodicity results to non-explicit invariant measures

Builds on functional inequalities for kinetic equations

Abstract

In non-equilibrium statistical physics models, the invariant measure $\mu$ of the process does not have an explicit density. In particular the adjoint $L^*$ in $L^2(\mu)$ of the generator $L$ is unknown and many classical techniques fail in this situation. An important progress has been made in [5] where functional inequalities are obtained for non-explicit steady states of kinetic equations under rather general conditions. However in [5] in the kinetic case the geometric ergodicity is only deduced from the functional inequalities for the case with conservative forces, corresponding to explicit steady states. In this note we obtain $L^2$ convergence rates in the non-equilibrium case.