Ergodicity of Langevin Dynamics and Its Discretizations for Nonsmooth Potentials with Linearly Growing Subgradient

This article is concerned with sampling from Gibbs distributions [Formula: see text] using Markov chain Monte Carlo methods. In particular, we investigate Langevin dynamics in the continuous- and discrete-time setting for such distributions with potentials [Formula: see text] which are strongly convex but possibly nondifferentiable and whose subgradients grow at most linearly. We show that the corresponding subgradient Langevin dynamics is exponentially ergodic with invariant distribution [Formula: see text] in the continuous setting and that its explicit discretization as well as a semi-implicit discretization are geometrically ergodic, the semi-implicit one exhibiting better convergence behavior for vanishing step sizes. We show that both discretizations approximate the target distribution [Formula: see text] for vanishing discretization step size. Moreover, we prove that the discrete schemes satisfy the law of large numbers allowing us to use consecutive iterates in order to compute statistics of the stationary distribution instead of simulating multiple independent Markov chains. Numerical experiments are provided confirming the theoretical findings and showcasing the practical relevance of the proposed methods in imaging applications.