Approximations of displacement interpolations by entropic interpolations

The Schrödinger problem is an entropy minimization problem on a set of path measures with prescribed initial and final marginals. It arises from a large deviation principle for the empirical measures of large particle systems. When the dynamics of the particles is slowed down while the prescribed marginals are unchanged, a second level large deviation phenomenon occurs. This leads to a sequence, indexed by the slowdown parameter, of Schrödinger problems which Gamma-converges to a dynamical optimal transport problem. We will illustrate these limits in the setting of L2 displacement interpolations on Rd and L1 displacement interpolations on graphs and Finsler manifolds.