Schrödinger Problem for Lattice Gases: A Heuristic Point of View

Aim of this paper is to take the first steps in the study of the Schrödinger problem for lattice gases (SPLG), which we formulate relying on classical results in large deviations theory. Our main contributions are a dynamical characterization of optimizers through a coupled system of PDEs and a precise study of the evolution and convexity of the quasi-potential along Schrödinger bridges. In particular, our computations show that, although SPLG does not admit a variational interpretation through Otto calculus, the fundamental geometric properties of the classical Schrödinger problem for independent particles still admit a natural generalization. These observations motivate the development of a Riemannian calculus on the space of probability measures associated with the class of geodesic distances studied in [3]. All our computations are formal, further efforts are needed to turn them into rigorous results.