Liouville Brownian motion

Abstract

We construct a stochastic process, called the Liouville Brownian motion, which is the Brownian motion associated to the metric $e^{γX(z)}\,dz^2$, $γ<γ_c=2$ 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 "$M_γ(dz)=e^{γX(z)}\,dz$". We prove that the associated Markov process is a Feller diffusion for all $γ<γ_c=2$ and that for all $γ<γ_c$, the Liouville measure $M_γ$ 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.