{
  "id": "1407.3240",
  "version": "v2",
  "published": "2014-07-11T18:17:58.000Z",
  "updated": "2015-01-30T15:24:17.000Z",
  "title": "Continuity and estimates of the Liouville heat kernel with applications to spectral dimensions",
  "authors": [
    "Sebastian Andres",
    "Naotaka Kajino"
  ],
  "comment": "34 pages; a number of small corrections have been made",
  "categories": [
    "math.PR"
  ],
  "abstract": "The Liouville Brownian motion (LBM), recently introduced by Garban, Rhodes and Vargas and in a weaker form also by Berestycki, is a diffusion process evolving in a planar random geometry induced by the Liouville measure $M_\\gamma$, formally written as $M_\\gamma(dz)=e^{\\gamma X(z)-{\\gamma^2} \\mathbb{E}[X(z)^2]/2} dz$, $\\gamma\\in(0,2)$, for a (massive) Gaussian free field $X$. It is an $M_\\gamma$-symmetric diffusion defined as the time change of the two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure $M_\\gamma$. In this paper we provide a detailed analysis of the heat kernel $p_t(x,y)$ of the LBM. Specifically, we prove its joint continuity, a locally uniform sub-Gaussian upper bound of the form $p_t(x,y)\\leq C_{1} t^{-1} \\log(t^{-1}) \\exp\\bigl(-C_{2}(|x-y|^{\\beta}/t)^{\\frac{1}{\\beta-1}}\\bigr)$ for small $t$ for each $\\beta>(\\gamma+2)^2/2$ and an on-diagonal lower bound of the form $p_{t}(x,x)\\geq C_{3}t^{-1}\\bigl(\\log(t^{-1})\\bigr)^{-\\eta}$ for small $t$ for $M_{\\gamma}$-a.e.\\ $x$ with some concrete constant $\\eta>0$. As applications, we also show that the pointwise spectral dimension equals $2$ $M_\\gamma$-a.e.\\ and that the global spectral dimension is $2$ as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-11T18:17:58.000Z",
      "abstract": "The Liouville Brownian motion, recently introduced by Garban, Rhodes and Vargas, is a diffusion process evolving in a planar random geometry induced by the Liouville measure $M_\\gamma$, formally written as $M_\\gamma(dz)=e^{\\gamma X(z)-{\\gamma^2} \\mathbb{E}[X(z)^2]/2} dz$, $\\gamma\\in(0,2)$, for a (massive) Gaussian free field $X$. It is an $M_\\gamma$-symmetric diffusion defined as the time-change of the standard two-dimensional Brownian motion by the positive continuous additive functional with Revuz measure $M_\\gamma$. In this paper we provide a detailed analysis of the heat kernel $p_t(x,y)$. Specifically, we prove its joint continuity, a locally uniform sub-Gaussian upper bound of the form $p_t(x,y)\\leq C_{1} t^{-1} \\log(t^{-1}) \\exp\\bigl(-C_{2}(|x-y|^{\\beta}/t)^{\\frac{1}{\\beta-1}}\\bigr)$ for small $t$ and an on-diagonal lower bound of the form $p_{t}(x,x)\\geq C_{3}t^{-1}\\bigl(\\log(t^{-1})\\bigr)^{-\\eta}$ for small $t$ for $M_{\\gamma}$-a.e. $x$ with some concrete constant $\\eta>0$. As applications, we also show that the pointwise spectral dimension equals $2$ $M_\\gamma$-a.e. and that the global spectral dimension is also $2$.",
      "comment": "33 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-01-30T15:24:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J35",
      "60J55",
      "60J60",
      "60K37",
      "31C25",
      "60J45",
      "60G15"
    ],
    "keywords": [
      "liouville heat kernel",
      "applications",
      "continuity",
      "standard two-dimensional brownian motion",
      "locally uniform sub-gaussian upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.3240A"
    }
  }
}