Long-time reverse transportation inequalities for non-globally-dissipative Langevin dynamics

TL;DR

This paper proves a dimension-free, uniform-in-time reverse transportation inequality for Langevin dynamics with non-convex potentials, extending long-time convergence results beyond log-concave cases.

Contribution

It introduces a novel reverse transportation inequality for non-convex Langevin dynamics that remains valid over long times and does not depend on dimension.

Findings

Establishes a uniform-in-time reverse transportation inequality for non-convex Langevin dynamics.

Shows exponential decay of the inequality in the long-time regime.

Extends results from log-concave to non-convex potential settings.

Abstract

We establish a dimension-free, uniform-in-time reverse transportation inequality for Langevin dynamics with non-convex potentials. This inequality controls the R\'enyi divergence of arbitrary order between the process distributions starting from distinct initial points and serves as the dual version of the Harnack inequality. Notably, we prove that this inequality retains exponential decay in the long-time regime, thereby extending existing results for log-concave sampling to the non-convex setting.