The authors consider the problem of minimizing the entropy of a law with respect to the law of a reference branching Brownian motion under density constraints at an initial and final time. They call this problem the branching Schrödinger problem by analogy with the Schrödinger problem, where the reference process is a Brownian motion. Whereas the Schrödinger problem is related to regularized (a.k.a. entropic) optimal transport, the authors investigate the link of the branching Schrödinger problem with regularized unbalanced optimal transport.

This link is shown at two levels. First, relying on duality arguments, the values of these two problems of calculus of variations are linked, in the sense that the value of the regularized unbalanced optimal transport (seen as a function of the initial and final measure) is the lower semi-continuous relaxation of the value of the branching Schrödinger problem. Second, the authors also explain a correspondence between the competitors of these two problems, and to that end they provide a fine description of laws having a finite entropy with respect to a reference branching Brownian motion.

The authors investigate the small noise limit, when the noise intensity of the branching Brownian motion goes to 0: in this case they show, at the level of the optimal transport model, that there is convergence to optimal partial transport. They also provide formal arguments about why looking at the branching Brownian motion, and not at other measure-valued branching Markov processes, like superprocesses, yields the problem closest to optimal transport. Finally, they explain how this problem can be solved numerically: the dynamical formulation of regularized unbalanced optimal transport can be discretized and solved via convex optimization.