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

Ioannis Karatzas; Bertram Tschiderer

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arXiv · 2010.04847

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

Ioannis Karatzas·Bertram Tschiderer

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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.

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Ioannis Karatzas, Bertram Tschiderer. 2020-10-09. A variational characterization of Langevin$\boldsymbol{-}$Smoluchowski diffusions. https://arxiv.org/abs/2010.04847

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