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Anomalous recurrence of Markov chains on negatively curved manifolds

Part of: Geometric probability and stochastic geometry Markov processes

Published online by Cambridge University Press:  06 October 2022

John Armstrong  and
Tim King Open the ORCID record for Tim King [Opens in a new window]
John Armstrong*
Affiliation:
King’s College London
Tim King*
Affiliation:
King’s College London
*
*Postal address: Department of Mathematics, Strand Building, Strand, London, WC2R 2LS
*Postal address: Department of Mathematics, Strand Building, Strand, London, WC2R 2LS
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Abstract

We present a recurrence–transience classification for discrete-time Markov chains on manifolds with negative curvature. Our classification depends only on geometric quantities associated to the increments of the chain, defined via the Riemannian exponential map. We deduce that a recurrent chain that has zero average drift at every point cannot be uniformly elliptic, unlike in the Euclidean case. We also give natural examples of zero-drift recurrent chains on negatively curved manifolds, including on a stochastically incomplete manifold.

Keywords

Non-homogeneous random walkuniform ellipticity

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 1 , March 2023 , pp. 204 - 222
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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