  In this paper we apply the 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.
