The most likely evolution of diffusing and vanishing particles: Schrodinger Bridges with unbalanced marginals

TL;DR

This paper generalizes the Schrödinger Bridge Problem to account for unbalanced marginals with particle losses, proposing a probabilistic model for the most likely evolution of diffusing and vanishing particles.

Contribution

It introduces a natural extension of SBP for lossy stochastic processes, incorporating particle killing and jump characteristics, which was not addressed in prior work.

Findings

Develops a Schrödinger Bridge framework for unbalanced marginals with losses.

Formulates a large-deviations approach to identify most probable particle evolution.

Provides a novel embedding for stochastic processes with diffusive and jump features.

Abstract

Stochastic flows of an advective-diffusive nature are ubiquitous in physical sciences. Of particular interest is the problem to reconcile observed marginal distributions with a given prior posed by E. Schrodinger in 1932/32 and known as the Schrodinger Bridge Problem (SBP). Due to its fundamental significance, interest in SBP has in recent years enticed a broad spectrum of disciplines. Yet, while the mathematics and applications of SBP have been developing at a considerable pace, accounting for marginals of unequal mass has received scant attention; the problem to interpolate between unbalanced marginals has been approached by introducing source/sink terms in an Adhoc manner. 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 Schrodinger's dictum; 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 law 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 stochastic evolution with losses, whereupon particles are "killed" according to a probabilistic law. 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, 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 Schrodinger's quest.