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

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.