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In this paper we apply the spectral theory of linear diffusions to study theone-dimensional Liouville Brownian Motion and Liouville Brownian excursionsfrom a given point. As an application we estimate the fractal dimensions oflevel sets of one-dimensional Liouville Brownian motion as well as variousprobabilistic asymptotic behaviours of Liouville Brownian motion and LiouvilleBrownian excursions.

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