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

Tony Lelièvre (1) , Mouad Ramil (1) , Julien Reygner (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

Consider the Langevin process, described by a vector (position,momentum) in Rd×Rd. Let O be a C2 open bounded and connected set of Rd. We prove the compactness of the semigroup of the Langevin process absorbed at the boundary of the domain D:=O×Rd. 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.

Domaines: Probabilités [math.PR]; Théorie spectrale [math.SP]

Tony Lelièvre, Mouad Ramil, Julien Reygner. Quasi-stationary distribution for the Langevin process in cylindrical domains, part I: existence, uniqueness and long-time convergence. 2021. ⟨hal-03123442v2⟩