{
  "id": "1202.1510",
  "version": "v4",
  "published": "2012-02-07T19:50:03.000Z",
  "updated": "2014-09-08T05:36:27.000Z",
  "title": "Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape",
  "authors": [
    "Georg Menz",
    "André Schlichting"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/14-AOP908 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2014, Vol. 42, No. 5, 1809-1884",
  "doi": "10.1214/14-AOP908",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "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\\\"{i} and Malrieu [Ann. Inst. Henri Poincar\\'{e} Probab. Stat. 46 (2010) 72-96]. The Eyring-Kramers formula follows as a simple corollary from two main ingredients: The first one shows that the PI and LSI constant of the diffusion restricted to metastable regions corresponding to the local minima scales well in $\\varepsilon$. This mimics the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of a mean-difference by a weighted transport distance. It contains the main contribution to the PI and LSI constant, resulting from exponentially long waiting times of jumps between metastable states of the diffusion.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-09-04T07:23:36.000Z",
      "abstract": "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 $L= \\varepsilon \\triangle - \\nabla H \\cdot \\nabla$ of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Westdickenberg and Villani; and of the mean-difference estimate introduced by Chafa\\\"i and Malrieu. The Eyring-Kramers formula follows as a simple corollary from two main ingredients: The first one shows that the PI and LSI constant of the diffusion restricted to metastable regions corresponding to the local minima scales well in $\\varepsilon$. This mimics the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of a mean-difference by a weighted transport distance. It contains the main contribution to the PI and LSI constant, resulting from exponentially long waiting times of jumps between metastable states of the diffusion.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2014-09-08T05:36:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J60",
      "35P15",
      "49R05"
    ],
    "keywords": [
      "logarithmic sobolev inequality",
      "energy landscape",
      "decomposition",
      "eyring-kramers formula",
      "lsi constant"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.1510M"
    }
  }
}