{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:24:26Z","timestamp":1787232266693,"version":"build-2736575974"},"reference-count":86,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100012025","name":"NYUAD","doi-asserted-by":"crossref","award":["76\/71260\/ADHPG\/AD356\/51119"],"award-info":[{"award-number":["76\/71260\/ADHPG\/AD356\/51119"]}],"id":[{"id":"10.13039\/100012025","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-24-1-0278"],"award-info":[{"award-number":["FA9550-24-1-0278"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-22-1-0292"],"award-info":[{"award-number":["W911NF-22-1-0292"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["ECCS-2347357"],"award-info":[{"award-number":["ECCS-2347357"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1942523"],"award-info":[{"award-number":["1942523"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[2025,8,7]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>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 \u201cdisappear\u201d 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. Schr\u00f6dinger in 1931\/32. It is now known as the Schr\u00f6dinger Bridge Problem (SBP). It turns out that Schr\u00f6dinger\u2019s 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 \u201cunbalanced\u201d 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 Schr\u00f6dinger\u2019s 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 \u201ckilled\u201d (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. Schr\u00f6dinger, 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\u2013Kac multiplicative reweighing of the reference measure. The latter, as we argue, is far from Schr\u00f6dinger\u2019s quest. An iterative scheme, generalizing the celebrated Fortet\u2013IPF\u2013Sinkhorn 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\u00a0of control variable. A numerical example illustrating the new theoretical results is also presented.<\/jats:p>","DOI":"10.1137\/25m176581x","type":"journal-article","created":{"date-parts":[[2025,8,7]],"date-time":"2025-08-07T07:16:50Z","timestamp":1754551010000},"page":"579-604","source":"Crossref","is-referenced-by-count":0,"title":["Optimal Survival Strategies for Diffusive Flows: A Schr\u00f6dinger Bridge Approach to Unbalanced Transport"],"prefix":"10.1137","volume":"67","author":[{"given":"Yongxin","family":"Chen","sequence":"first","affiliation":[{"name":"School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0012-5447","authenticated-orcid":true,"given":"Tryphon T.","family":"Georgiou","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Aerospace Engineering, University of California, Irvine, Irvine, CA 92697 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2648-2445","authenticated-orcid":true,"given":"Michele","family":"Pavon","sequence":"additional","affiliation":[{"name":"Division of Science, New York University, Abu Dhabi, U.A.E."},{"name":"Universit\u00e0 di Padova, Padova, 35121 Italy."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,8,7]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01222509"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1109\/LCSYS.2020.3047132"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.2307\/1970151"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316962"},{"key":"ref5","first-page":"219","volume":"85","author":"Birkhoff G.","year":"1957","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1007\/BF02169515"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.isci.2022.104987"},{"key":"ref8","first-page":"373","volume":"76","author":"Boltzmann L.","year":"1877","journal-title":"Wiener Berichte"},{"key":"ref9","unstructured":"C. Bunne, Neural Optimal Transport for Dynamical Systems Methods and Applications in Biomedicine, Dr.Sc. Thesis, ETH Z\u00fcrich, 2023."},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-6.2.256"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BF00247467"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2021.3060704"},{"key":"ref13","doi-asserted-by":"crossref","unstructured":"K. F. Caluya and A. Halder, Reflected Schr\u00f6dinger bridge: Density control with path constraints, in 2021 American Control Conference (ACC), IEEE, 2021, pp. 1137\u20131142.","DOI":"10.23919\/ACC50511.2021.9482813"},{"key":"ref14","unstructured":"T. Chen, G.H. Liu, and E. Theodorou, Likelihood training of Schr\u00f6dinger bridge using forward-backward SDEs theory, in International Conference on Learning Representations, preprint, arXiv:2110.11291."},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1137\/16M1061382"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2015.2457784"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2018.2791362"},{"key":"ref18","doi-asserted-by":"crossref","unstructured":"Y. Chen, T. T. Georgiou, and M. Pavon, Optimal steering of inertial particles diffusing anisotropically with losses, in Proc. American Control Conf., IEEE,\u00a02015, pp. 1252\u20131257.","DOI":"10.1109\/ACC.2015.7170905"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-015-0803-z"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-control-070220-100858"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1137\/20M1339982"},{"key":"ref22","unstructured":"Y. Chen, T. T. Georgiou, and M. Pavon, Control and Estimation of Multi-Commodity Network Flow under Aggregation, preprint, arXiv:2307.05103, 2023."},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1017\/S0956792518000219"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3303"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2018.03.008"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1007\/s10884-021-09977-1"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-018-0856-7"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1007\/BF01442404"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3462-3_55"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1063\/1.528840"},{"key":"ref31","first-page":"17695","volume":"34","author":"De Bortoli V.","year":"2021","journal-title":"Adv. Neural Inform. Process. Systems"},{"key":"ref32","series-title":"Stoch. Model. Appl. Probab 38","volume-title":"Large Deviations Techniques and Applications","author":"Dembo A.","year":"2009"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.24033\/bsmf.1494"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5208-5"},{"key":"ref35","doi-asserted-by":"crossref","unstructured":"A. Eldesoukey, M. Abdelgalil, and T. T. Georgiou, Collective Steering: Tracer-Informed Dynamics, preprint, arXiv:2505.01975, 2025.","DOI":"10.1109\/CDC57313.2025.11312474"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2024.3485537"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1007\/s11424-025-4440-9"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-018-1436-9"},{"key":"ref39","series-title":"Wiley Ser. Probab. Math. Statist. Probab. Math. Statist 282","volume-title":"Markov Processes: Characterization and Convergence","author":"Ethier S. N.","year":"2009"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0086180"},{"key":"ref41","first-page":"83","volume":"1","author":"Fortet R.","year":"1940","journal-title":"J. Math. Pure Appl. IX"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(89)90490-4"},{"key":"ref43","volume-title":"Partial Differential Equations of Parabolic Type","author":"Friedman A.","year":"2008"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-11079-5_2"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1063\/1.4915289"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2021.0148"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1007\/BF00532864"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.1007\/BF00535844"},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.103.012113"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-020-59045-9"},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1214\/23-AAP1969"},{"key":"ref52","unstructured":"C. L\u00e9onard, Stochastic Derivatives and Generalized H-Transforms of Markov Processes, preprint, arXiv:1102.3172, 2011."},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2011.11.026"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2014.34.1533"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.1214\/15-AOP1012"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-017-0599-0"},{"key":"ref57","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/ac85ea"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1007\/s00440-004-0340-4"},{"key":"ref59","doi-asserted-by":"publisher","DOI":"10.1137\/050631264"},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.106.044117"},{"key":"ref61","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.6.033070"},{"key":"ref62","doi-asserted-by":"publisher","DOI":"10.1063\/5.0256659"},{"key":"ref63","doi-asserted-by":"publisher","DOI":"10.1007\/BF00340014"},{"key":"ref64","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3458-6_9"},{"key":"ref65","volume-title":"Stochastic Differential Equations: An Introduction with Applications","author":"\u00d8ksendal B.","year":"2000","edition":"5"},{"key":"ref66","unstructured":"M. Pariset, Y.P. Hsieh, C. Bunne, A. Krause, and V. De Bortoli, Unbalanced diffusion Schr\u00f6dinger bridge, in ICML Workshop on New Frontiers in Learning, Control, and Dynamical Systems, preprint, arXiv:2306.09099, 2023."},{"key":"ref67","unstructured":"A. M. Pathan and M. Pavon, Entropy-Regularized Optimal Transport over Networks with Incomplete Marginals Information, preprint, arXiv:2404.00348, 2024."},{"key":"ref68","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.21975"},{"key":"ref69","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0443-5_22"},{"key":"ref70","volume-title":"Convergence of Stochastic Processes","author":"Pollard D.","year":"2012"},{"key":"ref71","series-title":"Grundlehren Math. Wiss. 293","volume-title":"Continuous Martingales and Brownian Motion","author":"Revuz D.","year":"2013"},{"key":"ref72","first-page":"11","volume":"42","author":"Sanov I. N.","year":"1957","journal-title":"Mat. Sb. (N.S.)"},{"key":"ref73","first-page":"144","author":"Schr\u00f6dinger E.","year":"1931","journal-title":"Sitzungsberichte der Preuss Akad. Wissen. Phys. Math. Klasse, Sonderausgabe"},{"key":"ref74","first-page":"269","volume":"3","author":"Schr\u00f6dinger E.","year":"1932","journal-title":"Annales de l\u2019institut Henri Poincar\u00e9, Presses Universitaires de France"},{"key":"ref75","unstructured":"R. Singh, I. Haasler, Q. Zhang, J. Karlsson, and Y. Chen, Inference with Aggregate Data: An Optimal Transport Approach, preprint, arXiv:2003.13933, 2020."},{"key":"ref76","unstructured":"V. R. Somnath, M. Pariset, Y.P. Hsieh, M. R. Martinez, A. Krause, and C. Bunne, Aligned diffusion Schr\u00f6dinger bridges, in Proceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence, Proc. Mach. Learn. Res. 216, R. J. Evans and I. Shpitser, eds. PMLR, 2023, pp. 1985\u20131995."},{"key":"ref77","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160190303"},{"key":"ref78","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970241"},{"key":"ref79","doi-asserted-by":"publisher","DOI":"10.3390\/e23091134"},{"key":"ref80","doi-asserted-by":"publisher","DOI":"10.1063\/1.528481"},{"key":"ref81","unstructured":"A. Wakolbinger, Schr\u00f6dinger bridges from 1931 to 1991, in Proc. 4th Latin American Congress in Probability and Mathematical Statistics, Mexico City, 1990, pp. 61\u201379."},{"key":"ref82","unstructured":"G. Wang, Y. Jiao, Q. Xu, Y. Wang, and C. Yang, Deep generative learning via Schr\u00f6dinger bridge, in International Conference on Machine Learning, PMLR, 2021, pp. 10794\u201310804."},{"key":"ref83","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.33.1532"},{"key":"ref84","first-page":"16280","volume":"34","author":"Zhang Q.","year":"2021","journal-title":"Adv. Neural Inform. Process. Syst."},{"key":"ref85","unstructured":"Q. Zhang and Y. Chen, Path integral sampler: A stochastic control approach for sampling, in International Conference on Learning Representations, 2022."},{"key":"ref86","unstructured":"Q. Zhang and Y. Chen, Fast sampling of diffusion models with exponential integrator, in Eleventh International Conference on Learning Representations, 2023."}],"container-title":["SIAM Review"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/25M176581X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:29:33Z","timestamp":1787228973000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/25M176581X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,7]]},"references-count":86,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,8,7]]}},"alternative-id":["10.1137\/25M176581X"],"URL":"https:\/\/doi.org\/10.1137\/25m176581x","relation":{},"ISSN":["0036-1445","1095-7200"],"issn-type":[{"value":"0036-1445","type":"print"},{"value":"1095-7200","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,7]]}}}