Convergence of unadjusted Hamiltonian Monte Carlo and Langevin dynamics via couplings

Abstract

In this talk, we analyse the long-time behaviour of two stochastic processes, namely the unadjusted Hamiltonian Monte Carlo method applied to mean-field particle models and the nonlinear Langevin dynamics of McKean-Vlasov type. We establish contraction bounds in L1 Wasserstein distance with dimension-free rates. In both cases, the results are not restricted to strongly convex confining potentials. The proofs are based on coupling approaches and on the construction of appropriate distance functions. For the Langevin dynamics, we additionally show uniform in time propagation of chaos bounds for the corresponding mean-field particle model.