Convergence of Reflected Langevin Diffusion for Constrained Sampling

TL;DR

This paper establishes a rigorous framework for approximating reflected Langevin diffusions in constrained domains, proving convergence of penalized SDEs and their discretizations to the true invariant measure.

Contribution

It introduces a sequence of penalized SDEs and proves their invariant measures converge to the reflected Langevin diffusion's measure, with explicit convergence rates.

Findings

Invariant measures of penalized SDEs converge in Wasserstein-2 distance.

Discretized penalized processes via Euler-Maruyama also converge.

Explicit polynomial rate of convergence established.

Abstract

We examine the Langevin diffusion confined to a closed, convex domain $D\subset\mathbb{R}^d$, represented as a reflected stochastic differential equation. We introduce a sequence of penalized stochastic differential equations and prove that their invariant measures converge, in Wasserstein-2 distance and with explicit polynomial rate, to the invariant measure of the reflected Langevin diffusion. We also analyze a time-discretization of the penalized process obtained via the Euler-Maruyama scheme and demonstrate the convergence to the original constrained measure. These results provide a rigorous approximation framework for reflected Langevin dynamics in both continuous and discrete time.