On Schrodinger bridges, entropic cost and their limits
Consider the Monge-Kantorovich problem of transporting densities $\rho_0$ to $\rho_1$ on $\mathbb{R}^d$ 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.