Spectral representation of one-dimensional Liouville Brownian motion and Liouville Brownian excursion

Abstract

In this paper we apply Krein's spectral theory of linear diffusions to study the one-dimensional Liouville Brownian motion and Liouville Brownian excursions from a given point. As an application we estimate the fractal dimensions of level sets of one-dimensional Liouville Brownian motion as well as various probabilistic asymptotic behaviours of Liouville Brownian motion and Liouville Brownian excursions.