Schrödinger Bridges for Systems of Interacting Particles

A Schrödinger bridge is the most probable time-dependent probability distribution that connects an initial probability distribution $w_i$ to a final one $w_f$. The problem has been solved and widely used for the case of simple Brownian evolution (non-interacting particles). It is related to the problem of entropy-regularized Wasserstein optimal transport. In this article, we generalize Brownian bridges to systems of interacting particles. We derive some equations for the forward and backward single particle ``wave-functions’’ which allow to compute the most probable evolution of the single-particle probability between the initial and final distributions.