Quasi-stationary distribution for the Langevin process in cylindrical domains, part I: existence, uniqueness and long-time convergence

TL;DR

This paper establishes the existence, uniqueness, and long-term convergence of the quasi-stationary distribution for the Langevin process in cylindrical domains, providing spectral insights and exponential convergence results.

Contribution

It proves the compactness of the semigroup, existence and uniqueness of the QSD, and spectral interpretation for the Langevin process in bounded cylindrical domains.

Findings

Existence and uniqueness of the QSD for the Langevin process.

Spectral characterization of the QSD.

Exponential convergence to the QSD conditioned on non-absorption.

Abstract

Consider the Langevin process, described by a vector (position,momentum) in $\mathbb{R}^{d}\times\mathbb{R}^d$. Let $\mathcal O$ be a $\mathcal{C}^2$ open bounded and connected set of $\mathbb{R}^d$. We prove the compactness of the semigroup of the Langevin process absorbed at the boundary of the domain $D:=\mathcal{O}\times\mathbb{R}^d$. We then obtain the existence of a unique quasi-stationary distribution (QSD) for the Langevin process on $D$. We also provide a spectral interpretation of this QSD and obtain an exponential convergence of the Langevin process conditioned on non-absorption towards the QSD.