{"paper":{"title":"Two-scale criteria for Poincar\\'{e} and log-Sobolev inequalities with applications to Markov chain Monte Carlo","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Vishwak Srinivasan","submitted_at":"2025-09-18T20:37:47Z","abstract_excerpt":"Given a collection of distributions $\\{P_{y}\\}$ and a mixing distribution $\\rho$ supported over $\\mathbb{R}^{d}$, we propose new sufficient conditions under which the mixture / joint distribution satisfies a Poincar\\'{e} or log-Sobolev inequality. We develop these sufficient conditions in a unified manner using the framework of $\\Phi$-Sobolev inequalities (Chafa\\\"{i}, 2004). The conditions that we develop in this work are satisfied by a variety of Markov chains, and consequently allows us to characterise the evolution of these functional inequalities for iterates generated by simulating these "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.15410","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.15410/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}