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Brownian motion, harmonic functions and hyperbolicity for Euclidean complexes

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Abstract.

A Euclidean complex X is a simplicial complex whose simplices are (flat) Euclidean simplices. We construct a natural Brownian motion on X and show that if X has nonpositive curvature and satisfies Gromov's hyperbolicity condition, then, with probability one, Brownian motion tends to a random limit on the Gromov boundary. Applying a combination of geometric and probabilistic techniques we describe spaces of harmonic functions on X.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD 20742, USA (e-mail: mbrin@math.umd.edu) , , , , , , US

    Michael Brin

  2. Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel (e-mail: kifer@math.huji.ac.il) , , , , , , IL

    Yuri Kifer

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  1. Michael Brin
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  2. Yuri Kifer
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Received November 18, 1999; in final form January 18, 2000 / Published online April 12, 2001

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Brin, M., Kifer, Y. Brownian motion, harmonic functions and hyperbolicity for Euclidean complexes. Math Z 237, 421–468 (2001). https://doi.org/10.1007/PL00004875

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  • DOI: https://doi.org/10.1007/PL00004875

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