Probability Seminar: Soumik Pal

Consider the Monge-Kantorovich problem of transporting densities $\rho_0$ to $\rho_1$ on $\rr^d$ with a strictly convex cost function. A popular relaxation of the problem is the one-parameter family called the entropic cost problem. The entropic cost J_h, h>0, is significantly faster to compute and $h J_h$ is known to converge to the optimal transport cost as $h$ goes to zero. We will give an overview of various ideas in this field, including discrete approximations, gamma convergence and particle systems. Finally we will discuss Gaussian approximations to Schrodinger bridges as $h$ approaches zero. As a consequence we obtain ``gradient flows’’ of entropy even in cases where the cost function is not a metric.