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

Consider the Langevin process which models the evolution of positions (in R-d) and associated momenta (in R-d) of interacting particles. Let O be a C-2 open bounded and connected set of R-d. We prove the compactness of the semigroup of the Langevin process absorbed at the boundary of the bounded-in-position domain D := O x R-d. We then obtain the existence of a unique quasi-stationary distribution (QSD) for the Langevin process on D. We provide a spectral interpretation of this QSD and obtain an exponential convergence of the Langevin process conditioned on non-absorption towards the QSD. We also give an explicit formula for the first exit point distribution from D, starting from the QSD.(C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This study investigates the properties of the Langevin process on a bounded-in-position domain, proving compactness of its semigroup and the existence of a unique quasi-stationary distribution. A spectral interpretation of the QSD is provided, along with exponential convergence of the process towards the QSD under non-absorption conditions. An explicit formula for the first exit point distribution from the domain, starting from the QSD, is given.