Diffusion Approximations to Schrödinger Bridges and the Convergence of Entropic Potentials

Consider the Monge-Kantorovich optimal transport problem between two Euclidean densities $μ$ and $ν$ and quadratic cost. The $\varepsilon$-Schrödinger bridge is the solution to the entropic regularized problem with regularization parameter $\varepsilon$. By taking the conditional expectation of the