Regularized unbalanced optimal transport and the large deviations of the branching Brownian motion

In the limit of small diffusivity, the large deviations of the space distribution of a large population of independent Brownian particles lead to quadratic optimal transport. This ten-year-old result by Léonard brought up to date the analysis of an entropic minimization problem introduced by Schrödinger in 1931, both for its relevance in applications and for its numerical properties. On the other hand, the large variety of natural phenomena exhibiting both transport and variations of mass - for instance in population dynamics - was a strong motivation for the development in the last few years of many so-called unbalanced optimal transport models, each of them proposing a quantitative compromise between these two behaviors. In this talk, based on a work in progress with Hugo Lavenant from Bocconi University, I will show how to derive a regularized version of an unbalanced optimal transport model from the large deviations of the branching Brownian motion. I will also present briefly how this study was motivated by a question in developmental biology.