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Anomalous recurrence of Markov chains on negatively curved manifolds

Part of: Markov processes Geometric probability and stochastic geometry

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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References

Anderson, J. W. (2005). Hyperbolic Geometry, 2nd edn. Springer, London.Google Scholar
Arnaudon, M., Barbaresco, F. and Yang, L. (2011). Medians and means in Riemannian geometry: Existence, uniqueness and computation. In Matrix Information Geometry, eds F. Nielsen and R. Bhatia. Springer, Berlin, pp. 169–197.Google Scholar
Cammarota, V. and Orsingher, E. (2008). Travelling randomly on the Poincaré half-plane with a Pythagorean compass. J. Statist. Phys. 130, 455–482.CrossRefGoogle Scholar
Carmo, M. P. do, (1992). Riemannian Geometry, 2nd edn. Birkhäuser, Boston, MA.CrossRefGoogle Scholar
Coulibaly-Pasquier, K. A. (2011). Brownian motion with respect to time-changing Riemannian metrics, applications to Ricci flow. Ann. Inst. Henri Poincaré Prob. Statist. 47, 515–538.CrossRefGoogle Scholar
Denisov, D., Korshunov, D. and Wachtel, V. (2016). At the edge of criticality: Markov chains with asymptotically zero drift. Preprint,arXiv:1612.01592.Google Scholar
Émery, M. and Mokobodzki, G. (1991). Sur le barycentre d’une probabilité dans une variété. Séminaire de probabilités de Strasbourg 25, 220–233.CrossRefGoogle Scholar
Georgiou, N., Menshikov, M. V., Mijatović, A. and Wade, A. R. (2016). Anomalous recurrence properties of many-dimensional zero-drift random walks. Adv. Appl. Prob. 48, 99–118.CrossRefGoogle Scholar
Georgiou, N., Mijatović, A. and Wade, A. R. (2019). Invariance principle for non-homogeneous random walks. Electron. J. Prob. 24, 1–38.CrossRefGoogle Scholar
Grigor’yan, A. (1999). Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. 36, 135–249.CrossRefGoogle Scholar
Hsu, E. P. (2002). Stochastic Analysis on Manifolds. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Ichihara, K. (1982). Curvature, geodesics and the Brownian motion on a Riemannian manifold (I): Recurrence properties. Nagoya Math. J. 87, 101–114.CrossRefGoogle Scholar
Jørgensen, E. (1975). The central limit problem for geodesic random walks. Z. Wahrscheinlichkeitsth. 32, 1–64.CrossRefGoogle Scholar
Jost, J. (2012). Nonpositive Curvature: Geometric and Analytic Aspects. Birkhäuser, Basel.Google Scholar
Kakutani, S. (1944). On Brownian motions in n-space. Proc. Imperial Acad. 20, 648–652.Google Scholar
Karlsson, A. (2004). Linear rate of escape and convergence in direction. In Random Walks and Geometry, ed. V. Kaimanovich. De Gruyter, Berlin, pp. 459–472.CrossRefGoogle Scholar
Kendall, W. S. (1984). Brownian motion on a surface of negative curvature. Séminaire de probabilités de Strasbourg 18, 70–76.Google Scholar
Kraaij, R. C., Redig, F. and Versendaal, R. (2019). Classical large deviation theorems on complete Riemannian manifolds. Stoch. Process. Appl. 129, 4294–4334.CrossRefGoogle Scholar
Lamperti, J. (1960). Criteria for the recurrence or transience of a stochastic process (I). J. Math. Anal. Appl. 1, 314–330.CrossRefGoogle Scholar
Lee, J. M. (1997). Riemannian Manifolds: An Introduction to Curvature. Springer, New York.CrossRefGoogle Scholar
Lenz, D., Sobieczky, F. and Woess, W. (2011). Random Walks, Boundaries and Spectra. Springer, Basel.CrossRefGoogle Scholar
Menshikov, M., Popov, S. and Wade, A. (2016). Non-homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems. Cambridge University Press.CrossRefGoogle Scholar
Menshikov, M. V. and Wade, A. R. (2010). Rate of escape and central limit theorem for the supercritical Lamperti problem. Stoch. Process. Appl. 120, 2078–2099.CrossRefGoogle Scholar
Paeng, S.-H. (2011). Brownian motion on manifolds with time-dependent metrics and stochastic completeness. J. Geom. Phys. 61, 940–946.CrossRefGoogle Scholar
Peres, Y., Popov, S. and Sousi, P. (2013). On recurrence and transience of self-interacting random walks. Bull. Brazil. Math. Soc. New Ser. 44, 841–867.CrossRefGoogle Scholar
Prohaska, R., Sert, C. and Shi, R. (2021). Expanding measures: Random walks and rigidity on homogeneous spaces. Preprint,arXiv:2104.09546.Google Scholar
Shiozawa, Y. (2017). Escape rate of the Brownian motions on hyperbolic spaces. Proc. Japan Acad. Ser. A 93, 27–29.CrossRefGoogle Scholar
Sturm, K.-T. (2002). Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature. Ann. Prob. 30, 1195–1222.CrossRefGoogle Scholar
Sullivan, D. (1983). The Dirichlet problem at infinity for a negatively curved manifold. J. Differential Geom. 18, 723–732.CrossRefGoogle Scholar
Williams, D. (1991). Probability with Martingales. Cambridge University Press.CrossRefGoogle Scholar