{"paper":{"title":"Explicit convergence rates of underdamped Langevin dynamics under weighted and weak Poincar\\'e--Lions inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","stat.CO"],"primary_cat":"math.PR","authors_text":"Andi Q. Wang, Gabriel Stoltz, Giovanni Brigati, Lihan Wang","submitted_at":"2024-07-22T20:19:51Z","abstract_excerpt":"We study the long-time behavior of the underdamped Langevin dynamics, in the case of so-called \\emph{weak confinement}. Indeed, any $\\mathrm{L}^\\infty$ distribution (in position and velocity) relaxes to equilibrium over time, and we quantify the convergence rate. In our situation, the spatial equilibrium distribution does not satisfy a Poincar\\'e inequality. Instead, we assume a weighted Poincar\\'e inequality, which allows for fat-tail or sub-exponential potential energies. We provide constructive and fully explicit estimates in $\\mathrm{L}^2$-norm for $\\mathrm{L}^\\infty$ initial data. A key-i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.16033","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/2407.16033/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"}