Besov spaces and random walks on a hyperbolic group: boundary traces and reflecting extensions of Dirichlet forms

Pierre Mathieu; Yuki Tokushige

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arXiv · 1812.01816

Besov spaces and random walks on a hyperbolic group: boundary traces and reflecting extensions of Dirichlet forms

Pierre Mathieu·Yuki Tokushige

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Abstract

We show the existence of a trace process at infinity for random walks on hyperbolic groups of conformal dimension < 2 and relate it to the existence of a reflecting random walk. To do so, we employ the theory of Dirichlet forms which connects the theory of symmetric Markov processes to functional analytic perspectives. We introduce a family of Besov spaces associated to random walks and prove that they are isomorphic to some of the Besov spaces constructed from the co-homology of the group studied in Bourdon-Pajot (2003). We also study the regularity of harmonic measures of random walks on hyperbolic groups using the potential theory associated to Dirichlet forms.

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Pierre Mathieu, Yuki Tokushige. 2018-12-05. Besov spaces and random walks on a hyperbolic group: boundary traces and reflecting extensions of Dirichlet forms. https://arxiv.org/abs/1812.01816

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