Liouville Brownian motion

TL;DR

This paper constructs the Liouville Brownian motion, a stochastic process linked to Liouville quantum gravity, by modifying standard Brownian motion with a measure derived from Gaussian Free Fields, and explores its properties.

Contribution

It introduces the Liouville Brownian motion, establishing its construction, Markov properties, and invariance of the Liouville measure, providing new tools for Liouville quantum gravity analysis.

Findings

Liouville Brownian motion is a Feller diffusion for b3<2.

Liouville measure is invariant under the process.

Constructs a framework for stochastic analysis in Liouville quantum gravity.

Abstract

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.