Abstract.

The interpretation of the Schrödinger problem as “noised” optimal transport is by now well established. From this perspective several natural questions stem, as for instance the convergence rate as the noise parameter vanishes of many quantities: optimal value, Schrödinger bridges and potentials… As for the optimal value, after the works of Erbar-Maas-Renger and Pal a first-order Taylor expansion is available. First aim of this talk is to improve this result in a twofold sense: from the first to the second order and from the Euclidean to the Riemannian setting (actually, to RCD spaces). From the proof it will be clear that the statement is in fact a particular instance of a more general result. For this reason, in the second part of the talk we introduce a large class of dynamical variational problems, extending far beyond the classical Schrödinger problem, and for them we prove $\Gamma$-convergence towards the geodesic problem and a suitable generalization of the second-order Taylor expansion.

(based on joint works with G. Conforti, L. Monsaingeon, and D. Vorotnikov)