A variational characterization of Langevin$\boldsymbol{-}$Smoluchowski diffusions

TL;DR

This paper presents a variational approach to Langevin–Smoluchowski diffusions, demonstrating invariance under time-reversal and linking stochastic control with Gibbs measures through entropy-based criteria.

Contribution

It introduces a novel entropic-type criterion for stochastic control of Langevin–Smoluchowski diffusions, connecting path space invariance with variational characterizations.

Findings

Invariance of measure under time-reversal

Convergence of control problems to Gibbs measure

Decreasing relative entropy along the flow

Abstract

We show that Langevin$-$Smoluchowski measure on path space is invariant under time-reversal, followed by stochastic control of the drift with a novel entropic-type criterion. Repeated application of these forward-backward steps leads to a sequence of stochastic control problems, whose initial/terminal distributions converge to the Gibbs probability measure of the diffusion, and whose values decrease to zero along the relative entropy of the Langevin$-$Smoluchowski flow with respect to this Gibbs measure.