Convergence rates for nonequilibrium Langevin dynamics

We study the exponential convergence to the stationary state for nonequilibrium Langevin dynamics, by a perturbative approach based on hypocoercive techniques developed for equilibrium Langevin dynamics. The Hamiltonian and overdamped limits (corresponding respectively to frictions going to zero or infinity) are carefully investigated. In particular, the maximal magnitude of admissible perturbations are quantified as a function of the friction. Numerical results based on a Galerkin discretization of the generator of the dynamics confirm the theoretical lower bounds on the spectral gap.RésuméNous considérons la convergence exponentielle vers l’état stationnaire pour des dynamiques de Langevin hors d’équilibre, par une approche perturbative reposant sur des techniques d’hypocoercivité initialement développées pour des dynamiques d’équilibre. Les limites hamiltoniennes et suramorties (qui correspondent respectivement au cas des frictions tendant vers zéro ou l’infini) sont étudiées précisément. En particulier, nous quantifions la magnitude maximale des perturbations admissibles en fonction de la friction. Des simulations numériques utilisant une discrétisation de Galerkin du générateur de la dynamique confirment les bornes inférieures que nous obtenons théoriquement pour le trou spectral.

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