Ergodic Foundations of Langevin-Based MCMC

In this work, we provide a comprehensive theoretical analysis of Langevin diffusion and its applications to Markov Chain Monte Carlo (MCMC) methods. We establish the ergodicity of continuous-time Langevin diffusion processes, proving their convergence to target distributions under suitable regularity conditions. The analysis is then extended to discrete-time settings, examining the properties of the Unadjusted Langevin Algorithm (ULA) and the Metropolis-Adjusted Langevin Algorithm (MALA). Employing tools from stochastic processes, ergodic theory, and Markov chain theory, we establish strong convergence results using Foster-Lyapunov drift conditions, coupling arguments, and geometric ergodicity. The paper explores connections between Langevin diffusion and optimal transport theory, highlighting recent developments in adaptive methods, transport map accelerated MCMC, and applications to high-dimensional Bayesian inference. Our theoretical results provide insights into algorithm design, parameter tuning, and convergence diagnostics for Langevin-based MCMC methods, bridging the gap between theory and practice in the development of efficient sampling algorithms for complex probability distributions.