The Annals of Probability

Liouville Brownian motion

Christophe Garban, Rémi Rhodes, and Vincent Vargas

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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.

Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 3076-3110.

Dates
Received: May 2014
Revised: June 2015
First available in Project Euclid: 2 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1470139160

Digital Object Identifier
doi:10.1214/15-AOP1042

Mathematical Reviews number (MathSciNet)
MR3531686

Zentralblatt MATH identifier
06631790

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 28A80: Fractals [See also 37Fxx]

Keywords
Liouville quantum gravity Liouville Brownian motion Gaussian multiplicative chaos

Citation

Garban, Christophe; Rhodes, Rémi; Vargas, Vincent. Liouville Brownian motion. Ann. Probab. 44 (2016), no. 4, 3076--3110. doi:10.1214/15-AOP1042. https://projecteuclid.org/euclid.aop/1470139160


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