Space-time divergence lemmas and optimal non-reversible lifts of diffusions on Riemannian manifolds with boundary

TL;DR

This paper develops a space-time divergence lemma with explicit constants to analyze non-reversible lifts of diffusions on Riemannian manifolds with boundary, demonstrating their optimal relaxation time reduction up to a constant factor.

Contribution

It extends previous divergence lemmas to general convex domains with boundary in Riemannian manifolds with lower curvature bounds, establishing the optimality of certain non-reversible lifts.

Findings

Non-reversible lifts can achieve relaxation times proportional to the domain diameter.

The divergence lemma provides explicit constants for bounds on relaxation times.

Optimal lifts balance deterministic transport and noise components.

Abstract

Non-reversible lifts reduce the relaxation time of reversible diffusions at most by a square root. For reversible diffusions on domains in Euclidean space, or, more generally, on a Riemannian manifold with boundary, non-reversible lifts are in particular given by the Hamiltonian flow on the tangent bundle, interspersed with random velocity refreshments, or perturbed by Ornstein-Uhlenbeck noise, and reflected at the boundary. In order to prove that for certain choices of parameters, these lifts achieve the optimal square-root reduction up to a constant factor, precise upper bounds on relaxation times are required. A key tool for deriving such bounds by space-time Poincar\'e inequalities is a quantitative space-time divergence lemma. Extending previous work of Cao, Lu and Wang, we establish such a divergence lemma with explicit constants for general locally convex domains with smooth boundary in Riemannian manifolds satisfying a lower, not necessarily positive, curvature bound. As a consequence, we prove optimality of the lifts described above up to a constant factor, provided the deterministic transport part of the dynamics and the noise are adequately balanced. Our results show for example that an integrated Ornstein-Uhlenbeck process on a locally convex domain with diameter $d$ achieves a relaxation time of the order $d$, whereas, in general, the Poincar\'e constant of the domain is of the order $d^2$.