The Annals of Probability

Liouville Brownian motion

Christophe Garban, Rémi Rhodes, and Vincent Vargas

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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

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

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

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Zentralblatt MATH identifier

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

Liouville quantum gravity Liouville Brownian motion Gaussian multiplicative chaos


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

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