{
  "id": "1410.3887",
  "version": "v1",
  "published": "2014-10-14T22:56:14.000Z",
  "updated": "2014-10-14T22:56:14.000Z",
  "title": "Regularization under diffusion and anti-concentration of temperature",
  "authors": [
    "Ronen Eldan",
    "James R. Lee"
  ],
  "categories": [
    "math.PR",
    "math.FA",
    "math.MG"
  ],
  "abstract": "Consider a non-negative function $f : \\mathbb R^n \\to \\mathbb R_+$ such that $\\int f\\,d\\gamma_n = 1$, where $\\gamma_n$ is the $n$-dimensional Gaussian measure. If $f$ is semi-log-convex, i.e. if there exists a number $\\beta \\geq 1$ such that for all $x \\in \\mathbb R^n$, the eigenvalues of $\\nabla^2 \\log f(x)$ are at least $-\\beta$, then $f$ satisfies an improved form of Markov's inequality: For all $\\alpha > e^3$, \\[ \\gamma_n(\\{x \\in \\mathbb R^n : f(x) > \\alpha \\}) \\leq \\frac{1}{\\alpha} \\cdot \\frac{C\\beta (\\log \\log \\alpha)^{4}}{\\sqrt{\\log \\alpha}}, \\] where $C$ is a universal constant. The bound is optimal up to a factor of $C \\sqrt{\\beta} (\\log \\log \\alpha)^{4}$, as it is met by translations and scalings of the standard Gaussian density. In particular, this implies that the mass on level sets of a probability density decays uniformly under the Ornstein-Uhlenbeck semigroup. This confirms positively the Gaussian case of Talagrand's convolution conjecture (1989).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-14T22:56:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regularization",
      "temperature",
      "anti-concentration",
      "dimensional gaussian measure",
      "standard gaussian density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.3887E"
    }
  }
}