Liouville Brownian motion

Abstract

We construct a stochastic process, called the Liouville Brownian motion, which is the Brownian motion associated to the metric $e^{\gamma X(z)}d{z}^2$, $\gamma <{\gamma }_c=2$ 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.

Published at http://dx.doi.org/10.1214/15-AOP1042 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• KPZ in one dimensional random geometry of multiplicative cascades
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• Hydrodynamic turbulence and intermittent random fields
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• Dynamical Freezing in a Spin Glass System with Logarithmic Correlations
• A Discovery Tour in Random Riemannian Geometry
• Invariance principles for random walks in random environment on trees
• Multifractal analysis of Gaussian multiplicative chaos and applications
• Construction of Liouville Brownian motion via Dirichlet form theory
• Conformally invariant random fields, quantum Liouville measures, and random Paneitz operators on Riemannian manifolds of even dimension
• Diffusion processes on branching Brownian motion