A Poincaré Inequality and Exponential Decay for the Elephant Random Walk

[2606.08884] A Poincaré Inequality and Exponential Decay for the Elephant Random Walk Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2606.08884 (math) [Submitted on 7 Jun 2026 ( v1 ), last revised 15 Aug 2026 (this version, v2)] Title: A Poincaré Inequality and Exponential Decay for the Elephant Random Walk Authors: Cristian Favio Coletti , Rafael de Mattos Grisi , Ioannis Papageorgiou View a PDF of the paper titled A Poincar\'e Inequality and Exponential Decay for the Elephant Random Walk, by Cristian Favio Coletti and 2 other authors View PDF HTML (experimental) Abstract: We study the long-time behaviour of a coninuous time one-dimensional elephant random walk with an absorbing boundary. By analyzing the associated evolution equation, we identify a proper limiting operator and establish a Poincaré inequality with spectral gap of order $N^{-2}$. As a consequence, we obtain matching exponential upper and lower bounds for the survival probability, showing that it decays at rate $e^{-ct/N^2}$. The proof relies on a decomposition of the generator into a limiting operator and a time-dependent perturbation, together with spectral estimates. Comments: 15 pages Subjects: Probability (math.PR) MSC classes: 60G50, 60J10, 35P15 Cite as: arXiv:2606.08884 [math.PR] (or arXiv:2606.08884v2 [math.PR] for this version) https://doi.org/10.48550/arXiv.2606.08884 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Ioannis Papageorgiou [ view email ] [v1] Sun, 7 Jun 2026 23:56:13 UTC (10 KB) [v2] Sat, 15 Aug 2026 02:02:31 UTC (11 KB) Full-text links: Access Paper: View a PDF of the paper titled A Poincar\'e Inequality and Exponential Decay for the Elephant Random Walk, by Cristian Favio Coletti and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2026-06 Change to browse by: math References & Citations NASA ADS Google 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? ) 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? ) 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) Major funding support from

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