{
  "id": "1301.2876",
  "version": "v4",
  "published": "2013-01-14T07:46:22.000Z",
  "updated": "2016-09-02T10:30:45.000Z",
  "title": "Liouville Brownian motion",
  "authors": [
    "Christophe Garban",
    "Rémi Rhodes",
    "Vincent Vargas"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/15-AOP1042 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2016, Vol. 44, No. 4, 3076-3110",
  "doi": "10.1214/15-AOP1042",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We construct a stochastic process, called the Liouville Brownian motion, which is the Brownian motion associated to the metric $e^{\\gamma X(z)}\\,dz^2$, $\\gamma<\\gamma_c=2$ and $X$ is a Gaussian Free Field. Such a process is conjectured to be related to the scaling limit of random walks on large planar maps eventually weighted by a model of statistical physics which are embedded in the Euclidean plane or in the sphere in a conformal manner. The construction amounts to changing the speed of a standard two-dimensional Brownian motion $B_t$ depending on the local behavior of the Liouville measure \"$M_{\\gamma}(dz)=e^{\\gamma X(z)}\\,dz$\". We prove that the associated Markov process is a Feller diffusion for all $\\gamma<\\gamma_c=2$ and that for all $\\gamma<\\gamma_c$, the Liouville measure $M_{\\gamma}$ is invariant under $P_{\\mathbf{t}}$. This Liouville Brownian motion enables us to introduce a whole set of tools of stochastic analysis in Liouville quantum gravity, which will be hopefully useful in analyzing the geometry of Liouville quantum gravity.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-06-02T19:49:49.000Z",
      "abstract": "We construct a stochastic process, called the {\\bf Liouville Brownian motion}, which is the Brownian motion associated to the metric $e^{\\gamma X(z)}dz^2$, $\\gamma <\\gamma_c=2$ and $X$ is a Gaussian Free Field. Such a process is conjectured to be the scaling limit of random walks on large planar maps eventually weighted by a model of statistical physics which are embedded in the Euclidean plane or in the sphere in a conformal manner. The construction amounts to changing the speed of a standard two-dimensional Brownian motion $B_t$ depending on the local behaviour of the Liouville measure \"$M_\\gamma(dz) = e^{\\gamma X(z)} dz$\". We prove that the associated Markov process is a Feller diffusion for all $\\gamma<\\gamma_c=2$ and that for all $\\gamma<\\gamma_c$, the Liouville measure $M_\\gamma$ is invariant under $P_\\t$. This Liouville Brownian motion enables us to introduce a whole set of tools of stochastic analysis in Liouville quantum gravity, which will be hopefully useful in analyzing the geometry of Liouville quantum gravity.",
      "comment": "29 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2016-09-02T10:30:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville quantum gravity",
      "planar maps",
      "standard two-dimensional brownian motion",
      "liouville brownian motion enables",
      "liouville measure"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1210730,
      "adsabs": "2013arXiv1301.2876G"
    }
  }
}