Entropic turnpike estimates for the kinetic Schr\"odinger problem

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.