Abstract
We investigate the kinetic Schrödinger problem, obtained considering Langevin dynamics instead of Brownian motion in Schrödinger’s thought experiment. Under a quasilinearity assumption we establish exponential entropic turnpike estimates for the corresponding Schrödinger 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 benefit from recent advances in the understanding of classical Schrödinger bridges and adaptations of Bakry–Émery 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 Schrödinger potentials.
Funding Statement
Giovanni Conforti acknowledges funding from the grant SPOT (ANR-20-CE40-0014). Giacomo Greco acknowledges support from NWO Research Project 613.009.111 “Analysis meets Stochastics: Scaling limits in complex systems”. This research was also partially funded by Nuffic in the framework of the Van Gogh Programme under the title “The kinetic Schrödinger Problem”.
Citation
Alberto Chiarini. Giovanni Conforti. Giacomo Greco. Zhenjie Ren. "Entropic turnpike estimates for the kinetic Schrödinger problem." Electron. J. Probab. 27 1 - 32, 2022. https://doi.org/10.1214/22-EJP850
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