Martin boundary of brownian motion on Gromov hyperbolic metric graphs

Soonki Hong; Seonhee Lim

doi:10.3934/dcds.2021014

Article Contents

2021, Volume 41, Issue 8: 3725-3757. Doi: 10.3934/dcds.2021014

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Martin boundary of brownian motion on Gromov hyperbolic metric graphs

Soonki Hong^1, , and
Seonhee Lim^1,2,

1.
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea
2.
Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea

^* Corresponding author: Soonki Hong
^* Corresponding author: Soonki Hong

Received: June 30, 2020

Early access: January 2021

Published: August 2021

This article based on Soonki Hong's Ph.D thesis

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Abstract

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 $.

Keywords:

General theory of random and stochastic dynamical systems,
Dirichlet forms,
Martin boundary theory.

Mathematics Subject Classification: Primary: 37H05; Secondary: 31C25, 31C35.

Citation:

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Figure 1. Tree approximation

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Figure 2. Ancona-Gouëzel inequality

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Figure 3. Strong Ancona-Gouëzel inequality

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Figure 4.

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