dc.contributor.author Garban, Christophe * dc.contributor.author Rhodes, Rémi * dc.contributor.author Vargas, Vincent * dc.date.accessioned 2013-01-16T14:10:25Z dc.date.available 2013-01-16T14:10:25Z dc.date.issued 2016 dc.identifier.issn 0091-1798 dc.identifier.uri https://basepub.dauphine.fr/handle/123456789/10847 dc.language.iso en en dc.subject Liouville quantum gravity dc.subject Random measures dc.subject Liouville Brownian motion dc.subject multiplicative chaos dc.subject.ddc 519 en dc.title Liouville Brownian motion dc.type Article accepté pour publication ou publié dc.contributor.editoruniversityother Unité de Mathématiques Pures et Appliquées (UMPA-ENSL) http://www.umpa.ens-lyon.fr/ CNRS : UMR5669 – École Normale Supérieure - Lyon;France dc.description.abstracten 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}). dc.relation.isversionofjnlname Annals of Probability dc.relation.isversionofjnlvol 44 dc.relation.isversionofjnlissue 4 dc.relation.isversionofjnldate 2016 dc.relation.isversionofjnlpages 3076-3110 dc.relation.isversionofdoi 10.1214/15-AOP1042 dc.identifier.urlsite https://arxiv.org/abs/1301.2876v4 dc.relation.isversionofjnlpublisher Institute of Mathematical Statistics dc.subject.ddclabel Probabilités et mathématiques appliquées en dc.description.submitted non en dc.description.ssrncandidate non dc.description.halcandidate non dc.description.readership recherche dc.description.audience International dc.relation.Isversionofjnlpeerreviewed oui dc.date.updated 2018-07-23T13:17:16Z hal.person.labIds * hal.person.labIds * hal.person.labIds *
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