The Sinkhorn or the Iterated Proportional Fitting Procedure (IPFP) algorithm is a fundamental

iterative algorithm with many uses in statistics and other areas of applications. For example, it

can be used to approximately compute an optimal transport coupling between two probability

measures. However, much of its behavior is still shrouded in mystery. We will talk about limit of

the iterates, as the temperature goes to zero, as an absolutely continuous curve on the

Wasserstein space that has three equivalent descriptions. One, it is a gradient flow of relative

entropy for a modified Wasserstein geometry. Two, it is the family of marginal distributions of a

novel family of diffusions. And, three, the family of measures can be generated by solutions of

the parabolic Monge-Ampere PDE. We will introduce this novel family of flows and related

stochastic processes and PDEs and talk about their properties including their rates to

equilibrium.

Based on joint work with Nabarun Deb, Young-Heon Kim and Geoff Schiebinger