Irreversible Langevin samplers and variance reduction: a large deviations approach

In order to sample from a given target distribution (often of Gibbs type), the Monte Carlo Markov chain method consists of constructing an ergodic Markov process whose invariant measure is the target distribution. By sampling the Markov process one can then compute, approximately, expectations of observables with respect to the target distribution. Often the Markov processes used in practice are time-reversible (i.e. they satisfy detailed balance), but our main goal here is to assess and quantify how the addition of a non-reversible part to the process can be used to improve the sampling properties. We focus on the diffusion setting (overdamped Langevin equations) where the drift consists of a gradient vector field as well as another drift which breaks the reversibility of the process but is chosen to preserve the Gibbs measure. In this paper we use the large deviation rate function for the empirical measure as a tool to analyze the speed of convergence to the invariant measure. We show that the addition of an irreversible drift leads to a larger rate function and it strictly improves the speed of convergence of ergodic average for (generic smooth) observables. We also deduce from this result that the asymptotic variance decreases under the addition of the irreversible drift and we give an explicit characterization of the observables whose variance is not reduced reduced, in terms of a nonlinear Poisson equation. Our theoretical results are illustrated and supplemented by numerical simulations.