Dynamical Schrödinger Bridge Problems on Graphs

We study a discrete dynamical Schrodinger bridge problem (SBP) as a dynamical variational problem on a finite graph. We prove that the discrete SBP exists a unique minimizer, which satisfies a boundary value Hamiltonian flow on probability simplex equipped with $$L^2$$ -Wasserstein metric. In our formulation, we establish the connection between discrete SBP problems and Hamiltonian flows.