  For a given target density, there exist an infinite number of diffusion
processes which are ergodic with respect to this density. As observed in a
number of papers, samplers based on nonreversible diffusion processes can
significantly outperform their reversible counterparts both in terms of
asymptotic variance and rate of convergence to equilibrium. In this paper, we
take advantage of this in order to construct efficient sampling algorithms
based on the Lie-Trotter decomposition of a nonreversible diffusion process
into reversible and nonreversible components. We show that samplers based on
this scheme can significantly outperform standard MCMC methods, at the cost of
introducing some controlled bias. In particular, we prove that numerical
integrators constructed according to this decomposition are geometrically
ergodic and characterise fully their asymptotic bias and variance, showing that
the sampler inherits the good mixing properties of the underlying nonreversible
diffusion. This is illustrated further with a number of numerical examples
ranging from highly correlated low dimensional distributions, to logistic
regression problems in high dimensions as well as inference for spatial models
with many latent variables.
