Couplings, gradient estimates and logarithmic Sobolev inequality for Langevin bridges

TL;DR

This paper provides new quantitative insights into Langevin bridges, establishing gradient estimates, couplings, comparison principles, and a logarithmic Sobolev inequality, along with convergence bounds for the invariant measure.

Contribution

It introduces a novel expression for the bridge drift using reciprocal characteristics, linking gradient estimates and couplings for Langevin bridges.

Findings

Established equivalence between gradient estimates and couplings.

Derived bounds for distances between different Langevin bridges.

Proved a logarithmic Sobolev inequality for bridge measures.

Abstract

In this paper we establish quantitative results about the bridges of the Langevin dynamics and the associated reciprocal processes. They include an equivalence between gradient estimates for bridge semigroups and couplings, comparison principles, bounds of the distance between bridges of different Langevin dynamics, and a Logarithmic Sobolev inequality for bridge measures. The existence of an invariant measure for the bridges is also discussed and quantitative bounds for the convergence to the invariant measure are proved . All results are based on a seemingly new expression of the drift of a bridge in terms of the reciprocal characteristic, which, roughly speaking, quantifies the " mean acceleration " of a bridge.