Macroscopic fluctuation theory from a Lagrangian viewpoint and the Schrödinger problem

We formulate the Schrödinger problem for interacting particle systems in the hydrodynamical regime thus extending the standard setting of independent particles. This involves the large deviations rate function for the empirical measure which is in fact a richer observable than the hydrodynamic observables density and current. In the case in which the constraints are the initial and final density, we characterize the optimal measure for the Schrödinger problem. We also introduce versions of the Schrödinger problem in which the constraints are related to the current and analyze the corresponding optimal measures.