  In order to sample from a given target distribution (often of Gibbs type),
the Monte Carlo Markov chain method consists in 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.
