# Liouville distorted Brownian motion

## TL;DR

This paper extends the concept of Liouville Brownian motion to a distorted Brownian motion setting, constructing the associated positive continuous additive functional with respect to the Liouville distorted measure.

## Contribution

It introduces the construction of the positive continuous additive functional for Liouville distorted Brownian motion starting from all points in 2, generalizing previous work on Liouville Brownian motion.

## Key findings

- Construction of the additive functional $(F_t)_{t \u2265 0}$ for Liouville distorted Brownian motion.
- Extension of Liouville Brownian motion framework to distorted Brownian motions.
- Theoretical foundation for further analysis of Liouville distorted measures.

## Abstract

The Liouville Brownian motion was introduced in \cite{GRV} as a time changed process $B_{A_t^{-1}}$ of a planar Brownian motion $(B_t)_{t \ge 0}$, where $(A_t)_{t \ge 0}$ is the positive continuous additive functional of $(B_t)_{t \ge 0}$ in the strict sense w.r.t. the Liouville measure. We first consider a distorted Brownian motion $(X_t)_{t\ge0}$ starting from all points in $\R^2$ associated to a Dirichlet form $(\E, D(\E))$ (see \cite{ShTr14}). We show that the positive continuous additive functional $(F_t)_{t \ge 0}$ of $(X_t)_{t \ge 0}$ in the strict sense w.r.t. the Liouville distorted measure can be constructed.