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

TL;DR

The paper addresses ergodicity and quantitative convergence for the kinetic Langevin process in non-equilibrium settings where the invariant measure $\mu$ has no explicit density. It combines functional inequalities from Huang-Kopfer-Ren with a modified-norm hypocoercivity approach to obtain explicit $L^2(\mu)$ convergence rates for the relative density $h_t=f_t/\mu$. The main result provides a rate $\|h_t-1\|_2 \le e^{-c \min(t,t^3)} \|h_0-1\|_2$ for some $c>0$, along with the deduction that $\mu$ satisfies a full-gradient Poincaré inequality, extending ergodicity results to non-equilibrium steady states. These findings offer a concrete and robust $L^2$-based convergence analysis in settings where $\mu$ is non-explicit and broadens the applicability of hypocoercivity techniques to kinetic models outside equilibrium.

Abstract

In non-equilibrium statistical physics models, the invariant measure $μ$ of the process does not have an explicit density. In particular the adjoint $L^*$ in $L^2(μ)$ 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.

Theorems & Definitions (7)

• Example 1
• Proposition 1
• Remark 1
• Theorem 2
• Proposition 3
• proof : Proof of Theorem \ref{['thm']}
• proof : Proof of Proposition \ref{['prop2']}