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Liouville Brownian motion

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  • معلومة اضافية
    • Contributors:
      Institut Camille Jordan (ICJ); École Centrale de Lyon (ECL); Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL); Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon); Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS); Probabilités, statistique, physique mathématique (PSPM); Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL); CEntre de REcherches en MAthématiques de la DEcision (CEREMADE); Université Paris Dauphine-PSL; Université Paris Sciences et Lettres (PSL)-Université Paris Sciences et Lettres (PSL)-Centre National de la Recherche Scientifique (CNRS); ANR-10-BLAN-0123,MAC2,Modèles aléatoires critiques bi-dimensionnels(2010); ANR-11-JS01-0005,CHAMU,Chaos multiplicatif gaussien(2011)
    • بيانات النشر:
      HAL CCSD
      Institute of Mathematical Statistics
    • الموضوع:
      2016
    • Collection:
      HAL Lyon 1 (University Claude Bernard Lyon 1)
    • نبذة مختصرة :
      International audience ; We construct a stochastic process, called the Liouville Brownian motion which we conjecture to be the scaling limit of random walks on large planar maps which are embedded in the euclidean plane or in the sphere in a conformal manner. Our construction works for all universality classes of planar maps satisfying $\gamma <\gamma_c=2$. In particular, this includes the interesting case of $\gamma=\sqrt{8/3}$ which corresponds to the conjectured scaling limit of large uniform planar $p$-angulations (with fixed $p\geq 3$). We start by constructing our process from some fixed point $x\in \R^2$ (or $x\in \S^2$). This amounts to changing the speed of a standard two-dimensional brownian motion $B_t$ depending on the local behaviour of the Liouville measure ''$M_\gamma(dz) = e^{\gamma X} dz$'' (where $X$ is a Gaussien Free Field, say on $\S^2$). A significant part of the paper focuses on extending this construction simultaneously to all points $x\in \R^2$ or $\S^2$ in such a way that one obtains a semi-group $P_\t$ (the Liouville semi-group). We prove that the associated Markov process is a Feller diffusion for all $\gamma<\gamma_c=2$ and that for $\gamma<\sqrt{2}$, the Liouville measure $M_\gamma$ is invariant under $P_\t$ (which in some sense shows that it is the right quantum gravity diffusion to consider). This Liouville Brownian motion enables us to give sense to part of the celebrated Feynman path integrals which are at the root of Liouville quantum gravity, the Liouville Brownian ones. Finally we believe that this work sheds some new light on the difficult problem of constructing a quantum metric out of the exponential of a Gaussian Free Field (see conjecture \ref{c.metric}).
    • Relation:
      info:eu-repo/semantics/altIdentifier/arxiv/1301.2876; hal-00773389; https://hal.science/hal-00773389; https://hal.science/hal-00773389/document; https://hal.science/hal-00773389/file/LBM-arxiv.pdf; ARXIV: 1301.2876
    • الدخول الالكتروني :
      https://hal.science/hal-00773389
      https://hal.science/hal-00773389/document
      https://hal.science/hal-00773389/file/LBM-arxiv.pdf
    • Rights:
      info:eu-repo/semantics/OpenAccess
    • الرقم المعرف:
      edsbas.B7F9B416