The dynamical Schrödinger problem in abstract metric spaces

In this paper we introduce the dynamical Schrödinger problem, defined for a wide class of entropy and Fisher information functionals, as a geometric problem on abstract metric spaces. Under very mild assumptions we prove a generic Γ-convergence result towards the geodesic problem as the noise parameter ε↓0. We also study the dependence of the entropic cost on the parameter ε. Some examples and applications are discussed.