Optimal Survival Strategies for Diffusive Flows: A Schrödinger Bridge Approach to Unbalanced Transport

Abstract

SIAM Review, Volume 67, Issue 3, Page 579-604, August 2025.
Abstract.Diffusive flows, and their discrete counterparts, are ubiquitous in the physical and engineering sciences. In many important examples, the total mass is not preserved and therefore standard probabilistic models are not suitable. Examples include electrons which may be absorbed by the medium in which they travel. In population genetics, some individuals may “disappear” due to their genotype. In traffic flows over a network, some vehicles might simply exit the circulation and park. In this more general situation, where some of the mass may be lost, it is of particular interest to reconcile the observed initial and final marginal distributions with a given prior. In the case when the two marginals are probability distributions, and thus of equal mass, this problem was posed and, to a considerable extent, solved by E. Schrödinger in 1931/32. It is now known as the Schrödinger Bridge Problem (SBP). It turns out that Schrödinger’s problem can be viewed as both a modeling and a control problem. Due to the fundamental significance of this problem, interest in the SBP and in its deterministic (zero-noise limit) counterpart of optimal mass transport (OMT) has in recent years enticed scientists from a broad spectrum of disciplines, including physics, stochastic control, computer science, probability theory, and geometry. Yet, while the mathematics and applications of SBP/OMT have been developing at a considerable pace, accounting for marginals of unequal mass has received scant attention. The problem of interpolating between “unbalanced” marginals has been approached by introducing source/sink terms into the transport equations in an ad hoc manner, chiefly driven by applications in image registration. Nevertheless, as hinted at above, losses are inherent in many physical processes and, thereby, models that account for lossy transport may also need to be reconciled with observed marginals following Schrödinger’s quest, that is, to adjust the probability of trajectories of particles, including those that do not make it to the terminal observation point, so that the updated evolution represents the most likely way that particles may have been transported, or vanished, at some intermediate point. Thus, the purpose of this work is to develop such a natural generalization of the SBP for diffusive evolution with losses, whereupon particles are “killed” (jump into a coffin/extinction state) according to a probabilistic law, and thereby mass is gradually lost along their stochastically driven flow. Through a suitable embedding, which appears to be novel, we turn the problem into an SBP for stochastic processes that combine diffusive and jump characteristics. Then, following a large-deviations formalism in the style of E. Schrödinger, given a prior law that allows for losses, we ask for the most probable evolution of particles along with the most likely killing rate as the particles transition between the specified marginals. Our approach differs sharply from previous work involving a Feynman–Kac multiplicative reweighing of the reference measure. The latter, as we argue, is far from Schrödinger’s quest. An iterative scheme, generalizing the celebrated Fortet–IPF–Sinkhorn algorithm, permits the computation of the new drift and the new killing rate of the path-space solution measure. We also formulate and solve a related fluid-dynamic control problem for the flow of one-time marginals where both the drift and the new killing rate play the role of control variable. A numerical example illustrating the new theoretical results is also presented.