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July 2016 Liouville Brownian motion
Christophe Garban, Rémi Rhodes, Vincent Vargas
Ann. Probab. 44(4): 3076-3110 (July 2016). DOI: 10.1214/15-AOP1042

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.

Citation

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Christophe Garban. Rémi Rhodes. Vincent Vargas. "Liouville Brownian motion." Ann. Probab. 44 (4) 3076 - 3110, July 2016. https://doi.org/10.1214/15-AOP1042

Information

Received: 1 May 2014; Revised: 1 June 2015; Published: July 2016
First available in Project Euclid: 2 August 2016

zbMATH: 06631790
MathSciNet: MR3531686
Digital Object Identifier: 10.1214/15-AOP1042

Subjects:
Primary: 28A80, 60D05

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 4 • July 2016
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