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Our results show that Schr\u00f6dinger bridges charge exponentially small masses outside the support of the limiting law that agrees with the optimal solution to the dynamical Monge\u2013Kantorovich optimal transport problem. Our proofs build on mixture representations of Schr\u00f6dinger bridges and establishing exponential continuity of Brownian bridges with respect to the initial and terminal points.<\/jats:p>","DOI":"10.1017\/jpr.2025.10049","type":"journal-article","created":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T08:13:57Z","timestamp":1765268037000},"page":"668-689","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["Large deviations for dynamical Schr\u00f6dinger problems"],"prefix":"10.1017","volume":"63","author":[{"given":"Kengo","family":"Kato","sequence":"first","affiliation":[{"name":"Cornell University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2025,12,9]]},"reference":[{"key":"S0021900225100491_ref26","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2011.11.026"},{"key":"S0021900225100491_ref16","unstructured":"[16] De Bortoli, V. , Thornton, J. , Heng, J. and Doucet, A. 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Published by Cambridge University Press on behalf of Applied Probability Trust","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}