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Biol."],"published-print":{"date-parts":[[2025,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Critical transitions and tipping phenomena between two meta\u2010stable states in stochastic dynamical systems are a scientific issue. In this work, we expand the methodology of identifying the most probable transition pathway between two meta\u2010stable states with Onsager\u2013Machlup action functional, to investigate the evolutionary transition dynamics between two meta\u2010stable invariant sets with Schr\u00f6dinger bridge. In contrast to existing methodologies such as statistical analysis, bifurcation theory, information theory, statistical physics, topology, and graph theory for early warning indicators, we introduce a novel framework on Early Warning Signals (EWS) within the realm of probability measures that align with the entropy production rate. To validate our framework, we apply it to the Morris\u2013Lecar model and investigate the transition dynamics between a meta\u2010stable state and a stable invariant set (the limit cycle or homoclinic orbit) under various conditions. Additionally, we analyze real Alzheimer\u2019s data from the Alzheimer\u2019s Disease Neuroimaging Initiative database to explore EWS indicating the transition from healthy to pre\u2010AD states. This framework not only expands the transition pathway to encompass measures between two specified densities on invariant sets, but also demonstrates the potential of our early warning indicators for complex diseases.<\/jats:p>","DOI":"10.1002\/qub2.86","type":"journal-article","created":{"date-parts":[[2025,3,13]],"date-time":"2025-03-13T04:19:14Z","timestamp":1741839554000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Action functional as an early warning indicator in the space of probability measures via Schr\u00f6dinger bridge"],"prefix":"10.1002","volume":"13","author":[{"given":"Peng","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics Huazhong University of Science and Technology  Wuhan China"},{"name":"Center for Mathematical Science Huazhong University of Science and Technology  Wuhan China"},{"name":"Steklov\u2010Wuhan Institute for Mathematical Exploration Huazhong University of Science and Technology  Wuhan China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7840-7060","authenticated-orcid":false,"given":"Ting","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics Huazhong University of Science and Technology  Wuhan China"},{"name":"Center for Mathematical Science Huazhong University of Science and Technology  Wuhan China"},{"name":"Steklov\u2010Wuhan Institute for Mathematical Exploration Huazhong University of Science and Technology  Wuhan China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jin","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics Huazhong University of Science and Technology  Wuhan China"},{"name":"Center for Mathematical Science Huazhong University of Science and Technology  Wuhan China"},{"name":"Steklov\u2010Wuhan Institute for Mathematical Exploration Huazhong University of Science and Technology  Wuhan China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jinqiao","family":"Duan","sequence":"additional","affiliation":[{"name":"Guangdong Provincial Key Laboratory of Mathematical and Neural Dynamical Systems Great Bay University  Dongguan China"},{"name":"Department of Mathematics and Department of Physics Great Bay University  Dongguan China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2025,3,12]]},"reference":[{"key":"e_1_2_10_2_1","unstructured":"DuanJ.An introduction to stochastic dynamics.Cambridge University Press. 2015."},{"key":"e_1_2_10_3_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0219493723400026"},{"issue":"5","key":"e_1_2_10_4_1","article-title":"An optimal control method to compute the most likely transition path for stochastic dynamical systems with jumps","volume":"32","author":"Wei W","year":"2022","journal-title":"Chaos: Interdiscipl J Nonlinear Sci"},{"key":"e_1_2_10_5_1","doi-asserted-by":"crossref","unstructured":"ChenJ GaoT LiY DuanJ.Detecting the most probable high dimensional transition pathway based on optimal control theory.2023. 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