Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T20:17:40.095Z Has data issue: false hasContentIssue false
  • English
  • Français

Spread out random walks on homogeneous spaces

Part of: Stochastic processes Probability theory on algebraic and topological structures Noncompact transformation groups Markov processes Ergodic theory

Published online by Cambridge University Press:  06 October 2020

ROLAND PROHASKA Open the ORCID record for ROLAND PROHASKA [Opens in a new window]
ROLAND PROHASKA*
Affiliation:
Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092Zürich, Switzerland
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A measure on a locally compact group is said to be spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the n-step distributions: they equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.

Keywords

Random walkhomogeneous spaceMarkov chainHarris recurrence

MSC classification

Primary: 37A50: Relations with probability theory and stochastic processes
Secondary: 22F30: Homogeneous spaces 60G50: Sums of independent random variables; random walks 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60J05: Discrete-time Markov processes on general state spaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3439 - 3473
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benoist, Y. and Quint, J.-F.. Mesures stationnaires et fermés invariants des espaces homogènes. Ann. Math. (2) 174(2) (2011), 1111–1162.CrossRefGoogle Scholar
Benoist, Y. and Quint, J.-F.. Introduction to random walks on homogeneous spaces. Jpn. J. Math. 7(2) (2012), 135–166.CrossRefGoogle Scholar
Benoist, Y. and Quint, J.-F.. Random walks on finite volume homogeneous spaces. Invent. Math. 187(1) (2012), 37–59.CrossRefGoogle Scholar
Benoist, Y. and Quint, J.-F.. Stationary measures and invariant subsets of homogeneous spaces (II). J. Amer. Math. Soc. 26(3) (2013), 659–734.CrossRefGoogle Scholar
Benoist, Y. and Quint, J.-F.. Stationary measures and invariant subsets of homogeneous spaces (III). Ann. Math. (2) 178(3) (2013), 1017–1059.CrossRefGoogle Scholar
Bourbaki, N.. Integration II. Chapters 7–9. Springer, Berlin, 2004. Translation of the 1963 and 1969 French originals.Google Scholar
Breuillard, E. F.. Equidistribution of random walks on nilpotent Lie groups and homogeneous spaces. PhD Thesis, Yale University, 2004.Google Scholar
Buenger, C. D.. Quantitative non-divergence, effective mixing, and random walks on homogeneous spaces. PhD Thesis, The Ohio State University, 2016.Google Scholar
Carne, T. K.. A transmutation formula for Markov chains. Bull. Sci. Math. (2) 109(4) (1985), 399–405.Google Scholar
Conze, J.-P. and Guivarc’h, Y.. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete Contin. Dyn. Syst. 33(9) (2013), 4239–4269.CrossRefGoogle Scholar
Eskin, A. and Margulis, G.. Recurrence properties of random walks on finite volume homogeneous manifolds. Random Walks and Geometry: Proceedings of a Workshop at the Erwin Schrödinger Institute (Vienna, June 18–July 13, 2001). Ed. Kaimanovich, V. A.. De Gruyter, Berlin, 2004, pp. 431–444. Corrected version: http://www.math.uchicago.edu/~eskin/return.ps.Google Scholar
Furman, A. and Shalom, Y.. Sharp ergodic theorems for group actions and strong ergodicity. Ergod. Th. & Dynam. Sys. 19(4) (1999), 1037–1061.CrossRefGoogle Scholar
Furstenberg, H.. Noncommuting random products. Trans. Amer. Math. Soc. 108(3) (1963), 377–428.CrossRefGoogle Scholar
Gallardo, L. and Schott, R.. Marches aléatoires sur les espaces homogènes de certains groupes de type rigide. Journées sur les marches aléatoires (Astérisque, 74). Société Mathématique de France, Paris, 1980, pp. 149–170.Google Scholar
Guivarc’h, Y. and Raja, C. R. E.. Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces. Ergod. Th. & Dynam. Sys. 32(4) (2012), 1313–1349.CrossRefGoogle Scholar
Hebisch, W. and Saloff-Coste, L.. Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21(2) (1993), 673–709.CrossRefGoogle Scholar
Hennion, H. and Roynette, B.. Un théorème de dichotomie pour une marche aléatoire sur un espace homogène. Journées sur les marches aléatoires (Astérisque, 74). Société Mathématique de France, Paris, 1980, pp. 99–122.Google Scholar
Kakutani, S.. Random ergodic theorems and Markoff processes with a stable distribution. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley, CA, 1951, pp. 247–261.Google Scholar
Levitan, M. L. and Smolowitz, L. H.. Limit theorems for reversible Markov processes. Ann. Probab. 1(6) (1973), 1014–1025.CrossRefGoogle Scholar
Lust-Piquard, F.. Lower bounds on $\parallel {K}^n{\parallel}_{1\to \infty }$ for some contractions K of ${L}^2\left(\mu \right)$ , with applications to Markov operators. Math. Ann. 303(4) (1995), 699–712.CrossRefGoogle Scholar
Margulis, G. A.. Discrete Subgroups of Semisimple Lie Groups. Springer, Berlin, 1991.CrossRefGoogle Scholar
Meyn, S. and Tweedie, R. L.. Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Nummelin, E.. Strong ratio limit theorems for $\phi$ -recurrent Markov chains. Ann. Probab. 7(4) (1979), 639–650.CrossRefGoogle Scholar
Orey, S.. Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold Co., London, 1971.Google Scholar
Raja, C. R. E. and Schott, R.. Recurrent random walks on homogeneous spaces of p-adic algebraic groups of polynomial growth. Arch. Math. (Basel) 91(4) (2008), 379–384.CrossRefGoogle Scholar
Revuz, D.. Markov Chains, 2nd edn. North-Holland, Publishing Co. Amsterdam, 1984.Google Scholar
Rosenthal, J. S.. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90(430) (1995), 558–566. Correction: J. Amer. Statist. Assoc. 90(431) (1995), 1136.CrossRefGoogle Scholar
Schott, R.. Irrfahrten auf nicht mittelbaren homogenen Räumen. Arbeitsber. Math. Inst. Univ. Salzburg 1981(1–2) (1981), 63–76.Google Scholar
Schott, R.. Marches aléatoires sur les espaces homogènes. Ann. Sci. Univ. Clermont-Ferrand II Math. 71(20) (1982), 131–139.Google Scholar
Schott, R.. Recurrent random walks on homogeneous spaces. Probability Measures on Groups, VIII (Oberwolfach, 1985). Springer, Berlin, 1986, pp. 146–152.CrossRefGoogle Scholar
Tuominen, P. and Tweedie, R. L.. Markov chains with continuous components. Proc. Lond. Math. Soc. (3) 38(1) (1979), 89–114.CrossRefGoogle Scholar