{"paper":{"title":"Poincar\\'{e} and logarithmic Sobolev inequalities by decomposition of the energy landscape","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.SP"],"primary_cat":"math.PR","authors_text":"Andr\\'e Schlichting, Georg Menz","submitted_at":"2012-02-07T19:50:03Z","abstract_excerpt":"We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian $H:\\mathbb {R}^n\\to \\mathbb {R}$ in the regime of low temperature $\\varepsilon$. We proof the Eyring-Kramers formula for the optimal constant in the Poincar\\'{e} (PI) and logarithmic Sobolev inequality (LSI) for the associated generator $L=\\varepsilon \\Delta -\\nabla H\\cdot\\nabla$ of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald et al. [Ann. Inst. Henri Poincar\\'{e} Probab. Stat. 45 (2009) 302-351] and of the mean-difference estimate introduced by Chafa\\\""},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1510","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}