Liouville Brownian motion
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Christophe Garban [2] ; Rémi Rhodes [1] ; Vincent Vargas [1]
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[1] Université Paris-Est
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[2] Université Lyon 1
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 4, 2016, págs. 3076-3110
- Idioma: inglés
- DOI: 10.1214/15-AOP1042
- Enlaces
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Resumen
We construct a stochastic process, called the Liouville Brownian motion, which is the Brownian motion associated to the metric eγX(z)dz2eγX(z)dz2, γ<γc=2γ<γc=2 and XX 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 BtBt depending on the local behavior of the Liouville measure “Mγ(dz)=eγX(z)dzMγ(dz)=eγX(z)dz”. We prove that the associated Markov process is a Feller diffusion for all γ<γc=2γ<γc=2 and that for all γ<γcγ<γc, the Liouville measure MγMγ is invariant under PtPt. 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.
