Entropic turnpike estimates for the kinetic Schrödinger problem

We investigate the kinetic Schrodinger problem, obtained considering Langevin dynamics instead of Brownian motion in Schrodinger's thought experiment. Under a quasilinearity assumption we establish exponential entropic turnpike estimates for the corresponding Schrodinger 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 Schrodinger 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 Schrodinger potentials.