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Martin boundary of Brownian motion on Gromov hyperbolic metric graphs. (English) Zbl 1471.37052

Summary: Let \(\widetilde{X}\) be a locally finite Gromov hyperbolic graph whose Gromov boundary consists of infinitely many points and with a cocompact isometric action of a discrete group \(\Gamma\). We show the uniform Ancona inequality for the Brownian motion which implies that the \(\lambda\)-Martin boundary coincides with the Gromov boundary for any \(\lambda\in[0,\lambda_0]\), in particular at the bottom of the spectrum \(\lambda_0\).

MSC:

37H05 General theory of random and stochastic dynamical systems
31C25 Dirichlet forms
31C35 Martin boundary theory
37E25 Dynamical systems involving maps of trees and graphs
60J65 Brownian motion
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