  In this paper, we analyse a proximal method based on the idea of
forward-backward splitting for sampling from distributions with densities that
are not necessarily smooth. In particular, we study the non-asymptotic
properties of the Euler-Maruyama discretization of the Langevin equation, where
the forward-backward envelope is used to deal with the non-smooth part of the
dynamics. An advantage of this envelope, when compared to widely-used
Moreu-Yoshida one and the MYULA algorithm, is that it maintains the MAP
estimator of the original non-smooth distribution. We also study a number of
numerical experiments that corroborate that support our theoretical findings.
