# Construction of Liouville Brownian motion via Dirichlet form theory

## TL;DR

This paper constructs the Liouville Brownian motion using Dirichlet form theory, establishing its foundation in Liouville quantum gravity by demonstrating measure smoothness and defining the process as a time change of planar Brownian motion.

## Contribution

It provides a rigorous construction of Liouville Brownian motion via Dirichlet forms, linking it to the Liouville measure and planar Brownian motion.

## Key findings

- Liouville measure shown to be smooth in the strict sense
- Liouville Brownian motion defined as a time change of planar Brownian motion
- Establishment of a rigorous mathematical framework for Liouville Brownian motion

## Abstract

The Liouville Brownian motion which was introduced in \cite{GRV} is a natural diffusion process associated with a random metric in two dimensional Liouville quantum gravity. In this paper we construct the Liouville Brownian motion via Dirichlet form theory. By showing that the Liouville measure is smooth in the strict sense, the positive continuous additive functional $(F_t)_{t \ge 0}$ of the Liouville measure in the strict sense w.r.t. the planar Brownian motion $(B_t)_{t \ge 0}$ is obtained. Then the Liouville Brownian motion can be defined as a time changed process of the planar Brownian motion $B_{F_t^{-1}}$.