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

Xiong Jin (University of Manchester)

Frank Adams 2,

The one-dimensional Liouville Brownian motion (LBM) under consideration is a generalized linear diffusion process with natural scale function and speed measure m, where m is the boundary Liouville measure on the real line obtained from the Gaussian free field on the half-plane with Neumann boundary condition. In this talk I will present some classical spectral representation of generalized linear diffusions and their excursions in terms of the speed measure m and its spectral measure. As an application the fractal dimension of the level sets of one-dimensional LBM as well as several probabilistic asymptotic behaviours of LBM are deduced.

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