THE MOST LIKELY EVOLUTION OF DIFFUSING AND VANISHING PARTICLES: SCHRÖDINGER BRIDGES WITH UNBALANCED MARGINALS

Stochastic flows of an advective-diffusive nature are ubiquitous in biology and the physical sciences. Of particular interest is the problem of reconciling observed marginal distributions with a given prior posed by Schrödinger in 1932 and known as the Schrödinger Bridge Problem (SBP). It turns out that Schrödinger's problem can be viewed as both a modeling problem and a control problem. Due to their fundamental significance, the SBP and its deterministic (zero-noise limit) counterpart of Optimal Mass Transport (OMT) have recently received interest within a broad spectrum of disciplines, including physics, stochastic control, computer science, probability theory, and geometry. Yet, while the mathematics and applications of the 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, 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 dictum; that is, it is necessary to adjust the probability of trajectories of particles, including those that do not make it to the terminal observation point, so that the updated law represents the most likely way that particles may have been transported, or have vanished, at some intermediate point. Thus, the purpose of this work is to develop such a natural generalization of the SBP for stochastic 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 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 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 us to compute the new drift and the new killing rate of the path-space solution measure. We finally 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 variables.