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

arXiv: https://arxiv.org/abs/2411.12051 · HTML: https://mlsanity.com/paper/2411.12051/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Nov 2024.

## Built on by (most cited first, 3 of 3)

- [Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]](https://mlsanity.com/paper/2502.01358/) (2025)
- [The Performance Of The Unadjusted Langevin Algorithm Without Smoothness Assumptions](https://mlsanity.com/paper/2502.03458/) (2025)
- [The Tamed Subgradient Unadjusted Langevin Algorithm beyond Convexity](https://mlsanity.com/paper/2608.06283/) (2026)

## Abstract (arXiv)

This article is concerned with sampling from Gibbs distributions π(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 π in the continuous setting and that certain explicit as well as semi-implicit discretizations are geometrically ergodic and approximate π 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.