Abstract

Schrödinger’s classical thought experiment aimed for finding the most likely evolution between two subsequent observations of a cloud of independent particles. We introduce and advocate a generalised Schrödinger problem, defined for a wide class of entropy and Fisher information functionals, as a geometric problem on a metric space. Broadly speaking, in Schrödinger’s original situation the metric space is the Wasserstein space and the entropy is Boltzmann’s one. Under very mild assumptions (in particular, without any curvature restrictions) we prove a generic gamma-convergence result of the generalised Schrödinger problem towards the geodesic problem, as the temperature parameter tends to zero. Our novel technique is based on adaptive perturbations by gradient flows. We then study the dependence of the entropic cost on the temperature parameter. A similar technique allows us to prove the so-called Ishihara property for the harmonic maps valued in metric spaces, which means that the pullback of a convex function defined on a metric space by a harmonic map valued in that space is subharmonic. This abstract result implies some conjectures of [Y. Brenier, “Extended Monge-Kantorovich theory” in Optimal transportation and applications (Martina Franca, 2001), volume 1813 of Lecture Notes in Math., pages 91-121. Springer, Berlin, 2003]. The talk will be based on joint works with H. Lavenant, L. Monsaingeon & L. Tamanini.