{
  "id": "1410.6922",
  "version": "v1",
  "published": "2014-10-25T14:02:04.000Z",
  "updated": "2014-10-25T14:02:04.000Z",
  "title": "Quantitative logarithmic Sobolev inequalities and stability estimates",
  "authors": [
    "Max Fathi",
    "Emanuel Indrei",
    "Michel Ledoux"
  ],
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "We establish an improved form of the classical logarithmic Sobolev inequality for the Gaussian measure restricted to probability densities which satisfy a Poincar\\'e inequality. The result implies a lower bound on the deficit in terms of the quadratic Kantorovich-Wasserstein distance. We similarly investigate the deficit in the Talagrand quadratic transportation cost inequality this time by means of an ${\\rm L}^1$-Kantorovich-Wasserstein distance, optimal for product measures, and deduce a lower bound on the deficit in the logarithmic Sobolev inequality in terms of this metric. Applications are given in the context of the Bakry-\\'Emery theory and the coherent state transform. The proofs combine tools from semigroup and heat kernel theory and optimal mass transportation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-25T14:02:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "logarithmic sobolev inequality",
      "quantitative logarithmic sobolev inequalities",
      "stability estimates",
      "talagrand quadratic transportation cost inequality",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.6922F"
    }
  }
}