Stability of Partially Implicit Langevin Schemes and Their MCMC Variants

A broad class of implicit or partially implicit time discretizations for the Langevin diffusion are considered and used as proposals for the Metropolis–Hastings algorithm. Ergodic properties of our proposed schemes are studied. We show that introducing implicitness in the discretization leads to a process that often inherits the convergence rate of the continuous time process. These contrast with the behavior of the naive or Euler–Maruyama discretization, which can behave badly even in simple cases. We also show that our proposed chains, when used as proposals for the Metropolis–Hastings algorithm, preserve geometric ergodicity of their implicit Langevin schemes and thus behave better than the local linearization of the Langevin diffusion. We illustrate the behavior of our proposed schemes with examples. Our results are described in detail in one dimension only, although extensions to higher dimensions are also described and illustrated.