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

Abstract

Stochastic flows of an advective-diffusive nature are ubiquitous in biology and the physical sciences. Of particular interest is the problem to reconcile observed marginal distributions with a given prior posed by E. Schrödinger in 1932/32 and known as the Schrödinger Bridge Problem (SBP). It turns out that Schrödinger's problem can be viewed both as a modeling as well as a control problem. Due to the fundamental significance of this problem, interest in SBP and in its deterministic (zero-noise limit) counterpart of Optimal Mass Transport (OMT) has in recent years enticed 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 to interpolate between "unbalanced" marginals has been approached by introducing source/sink terms into the transport equations, in an adhoc 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, to adjust the probabilty 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" (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 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 to compute the new drift and the new killing rate of the path-space solution measure. We finally formulate and solve a related fluiddynamic control problem for the flow of one-time marginals were both the drift and the new killing rate play the role of control variables.

Regularized unbalanced optimal transport as entropy minimization with respect to branching Brownian motion

“…In the second case, it is the other way around: Ψ 1,p is infinite on R * + (Figure A.6), and as the reference BBM cannot create new particles, it is forbidden to consider competitors of the corresponding RUOT problem which gain mass. In this second case, taking p 0 = 1, we can actually carry explicit computations: we find Ψ 1,p = l(|r|) = |r| log |r| − |r| + 1 on R − , which coincides with formula (33) in [CGP21].…”

“…One can find a similar formulation in [CGP21] where the penalization Ψ prevents r from taking positive values: only mass removal is allowed.…”

“…First, we give a probabilistic justification for models of unbalanced optimal transport, which has actually been an open question since their introduction. As far as we know, the recent paper [CGP21] is the only other work treating this question, and it is done by looking at processes along which particles may be killed: from the unbalanced transport point of view, mass can be removed but never added. On the other hand in the present work both are possible.…”

Conditioning diffusion processes with respect to the local time at the origin

“…Among the different conditioned diffusions that have been constructed besides the basic example of the Brownian Bridge, one can cite the Brownian excursion [11,12], the Brownian meander [13], the taboo processes [14][15][16][17][18][19], or non-intersecting Brownian bridges [20]. Let us also mention the conditioning in the presence of killing rates [3,[21][22][23][24][25][26][27][28] or when the killing occurs only via an absorbing boundary condition [29][30][31][32]. Note that stochastic bridges have been studied for many other Markov processes, including various diffusions processes [33][34][35], discretetime random walks and Lévy flights [36][37][38], continuous-time Markov jump processes [38], run-and-tumble trajectories [39], or processes with resetting [40].…”

Conditioning diffusion processes with respect to the local time at the origin

“…Among the different conditioned diffusions that have been constructed besides the basic example of the Brownian Bridge, one can cite the Brownian excursion [11,12], the Brownian meander [13], the taboo processes [14][15][16][17][18][19], or non-intersecting Brownian bridges [20]. Let us also mention the conditioning in the presence of killing rates [3,[21][22][23][24][25][26][27][28] or when the killing occurs only via an absorbing boundary condition [29][30][31][32]. Note that stochastic bridges have been studied for many other Markov processes, including various diffusions processes [33][34][35], discrete-time random walks and Lévy flights [36][37][38], continuous-time Markov jump processes [38], run-and-tumble trajectories [39], or processes with resetting [40].…”