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 C² 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×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.