Entropic turnpike estimates for the kinetic Schr\"odinger problem

TL;DR

This paper extends the Schr"odinger problem to kinetic Langevin dynamics, establishing exponential entropic turnpike estimates and convergence results, thereby advancing understanding of kinetic Schr"odinger bridges and related inequalities.

Contribution

It introduces entropic turnpike estimates for the kinetic Schr"odinger problem under quasilinearity, connecting recent advances in classical Schr"odinger bridges to the kinetic setting.

Findings

Exponential entropic turnpike estimates established.

Fast convergence of entropic cost to marginal entropies.

Derivation of an entropic Talagrand inequality.

Abstract

We investigate the kinetic Schr\"odinger problem, obtained considering Langevin dynamics instead of Brownian motion in Schr\"odinger's thought experiment. Under a quasilinearity assumption we establish exponential entropic turnpike estimates for the corresponding Schr\"odinger bridges and exponentially fast convergence of the entropic cost to the sum of the marginal entropies in the long-time regime, which provides as a corollary an entropic Talagrand inequality. In order to do so, we profit from recent advances in the understanding of classical Schr\"odinger bridges and adaptations of Bakry-\'Emery formalism to the kinetic setting. Our quantitative results are complemented by basic structural results such as dual representation of the entropic cost and the existence of Schr\"odinger potentials.