Resumen de Liouville Brownian motion

Christophe Garban, Rémi Rhodes, Vincent Vargas

• 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.