Beyond the Arcsine Law: Exact Two-Time Statistics of the Occupation Time in Jump Processes

[2510.25859] Beyond the Arcsine Law: Exact Two-Time Statistics of the Occupation Time in Jump Processes Skip to main content Search Submit Donate Log in Condensed Matter > Statistical Mechanics arXiv:2510.25859 (cond-mat) [Submitted on 29 Oct 2025] Title: Beyond the Arcsine Law: Exact Two-Time Statistics of the Occupation Time in Jump Processes Authors: Arthur Plaud , Olivier Bénichou View PDF HTML (experimental) Abstract: Occupation times quantify how long a stochastic process remains in a region, and their single-time statistics are famously given by the arcsine law for Brownian and Lévy processes. By contrast, two-time occupation statistics, which directly probe temporal correlations and aging, have resisted exact characterization beyond renewal processes. In this Letter we derive exact results for generic one-dimensional jump processes, a central framework for intermittent and discretely sampled dynamics. Using generalized Wiener-Hopf methods, we obtain the joint distribution of occupation time and position, the aged occupation-time law, and the autocorrelation function. In the continuous-time scaling limit, universal features emerge that depend only on the tail of the jump distribution, providing a starting point for exploring aging transport in complex environments. Comments: 6 pages + 32 pages of supplementary material Subjects: Statistical Mechanics (cond-mat.stat-mech) ; Probability (math.PR) Cite as: arXiv:2510.25859 [cond-mat.stat-mech] (or arXiv:2510.25859v1 [cond-mat.stat-mech] for this version) https://doi.org/10.48550/arXiv.2510.25859 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Phys. Rev. E 114, 014137 (2026) Related DOI : https://doi.org/10.1103/ggwv-ckyj Focus to learn more DOI(s) linking to related resources Submission history From: Arthur Plaud [ view email ] [v1] Wed, 29 Oct 2025 18:03:39 UTC (1,013 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: cond-mat.stat-mech < prev | next > new | recent | 2025-10 Change to browse by: cond-mat math math.PR References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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? ) 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? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) 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? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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