A probabilistic view on unbalanced optimal transport

Optimal transport asks the question: what is the optimal way to transport a distribution of mass from one configuration to another. One of its variant, regularized optimal transport, is closely connect to large deviation principle and entropy minimization with respect to the law of the Brownian motion (a.k.a. the Schrödinger problem). In short: regularized optimal transport has a neat and fruitful probabilistic interpretation. I will explain what happens when we replace Brownian motion by branching Brownian motion (that is, when particles may also split or die at random instants): the optimal transport counterpart becomes regularized unbalanced optimal transport, enabling to match distributions of unequal mass.

This is joint work with Aymeric Baradat, see Arxiv preprint 2111.01666.