Ergodicity of Langevin Dynamics and its Discretizations for Non-smooth Potentials

This article is concerned with sampling from Gibbs distributions $\pi(x)\propto e^{-U(x)}$ using Markov chain Monte Carlo methods. In particular, we investigate Langevin dynamics in the continuous- and the discrete-time setting for such distributions with potentials $U(x)$ which are strongly-convex but possibly non-differentiable. We show that the corresponding subgradient Langevin dynamics are exponentially ergodic to the target density $\pi$ in the continuous setting and that certain explicit as well as semi-implicit discretizations are geometrically ergodic and approximate $\pi$ for vanishing discretization step size. Moreover, we prove that the discrete schemes satisfy the law of large numbers allowing to use consecutive iterates of a Markov chain in order to compute statistics of the stationary distribution posing a significant reduction of computational complexity in practice. Numerical experiments are provided confirming the theoretical findings and showcasing the practical relevance of the proposed methods in imaging applications.