Construction of Liouville Brownian motion via Dirichlet form theory

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 $B_{F_t^{-1}}$.

accepted

References in corpus (2)

• On the stochastic regularity of distorted Brownian motions
• Elliptic regularity results: n-regularized Liouville Brownian motion and non-symmetric diffusions associated with degenerate forms