Trajectory of Probabilities, Probability on Trajectories, and the Stochastic-Quantum Correspondence
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Quantum Physics arXiv:2602.23491 (quant-ph) [Submitted on 26 Feb 2026] Title:Trajectory of Probabilities, Probability on Trajectories, and the Stochastic-Quantum Correspondence Authors:Győző Egri, Marton Gomori, Balazs Gyenis, Gábor Hofer-Szabó View a PDF of the paper titled Trajectory of Probabilities, Probability on Trajectories, and the Stochastic-Quantum Correspondence, by Gy\H{o}z\H{o} Egri and 3 other authors View PDF HTML (experimental) Abstract:The probabilistic description of the time evolution of a physical system can take two conceptually distinct forms: a trajectory of probabilities, which specifies how probabilities evolve over time, and a probability on trajectories, which assigns probabilities to possible histories. A lack of a clear distinction between these two probabilistic descriptions has given rise to a number of conceptual difficulties, particularly in recent analyses of stochastic-quantum correspondence. This paper provides a systematic account of their relationship. We define probability dynamics and stochastic process families together with a precise notion of implementation that connects the two descriptions. We show that implementations are generically non-unique, that every probability dynamics admits a Markovian implementation, and characterize when non-Markovian implementations are possible. We expose fallacies in common arguments for the linearity of probability dynamics based on the law of total probability and clarify the proper interpretation of ``transition matrices'' by distinguishing dynamics-level maps from the conditional probability matrices of implementing processes. We further introduce decomposability as the appropriate general notion of stepwise evolution for (possibly nonlinear) probability dynamics, relate it to divisibility in the linear case -- showing that the two can come apart -- and disentangle both notions from Markovianity and time-homogeneity. Finally, we connect these results to what we call statistical dynamics, in which linearity is indeed physically motivated, and contrast the framework with quantum mechanics. Subjects: Quantum Physics (quant-ph); History and Philosophy of Physics (physics.hist-ph) Cite as: arXiv:2602.23491 [quant-ph] (or arXiv:2602.23491v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2602.23491 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Balazs Gyenis [view email] [v1] Thu, 26 Feb 2026 20:53:16 UTC (66 KB) Full-text links: Access Paper: View a PDF of the paper titled Trajectory of Probabilities, Probability on Trajectories, and the Stochastic-Quantum Correspondence, by Gy\H{o}z\H{o} Egri and 3 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-02 Change to browse by: physics physics.hist-ph References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)