The historical Schrödinger problem (~1930) consists in reconstructing the most likely trajectory of a system of particles, given the observation of its statistical distribution at two initial and terminal times. Recently, deep links with optimal transport were discovered, allowing to view the the Schrödinger problem as a noisy version of the geodesic problem in the Wasserstein space of probability measures. The level of noise is determined by a small temperature $\varepsilon>0$ and is driven by the Boltzmann entropy. In the small noise limit, it is known that the blurred problem Gamma-converges towards the deterministic one, which is actually remarkably useful for numerics. In this talk I will discuss a natural extension to geometric Schrödinger problems driven by general entropy functionals on arbitrary metric spaces, for which a general Gamma-convergence results holds and connections with geodesic convexity can be established.
This is based on joint works with L. Tamanini (CEREMADE/INRIA-Mokaplan) and D. Vorotnikov (Univ. Coimbra).