Hostname: page-component-76d6cb85b7-jhrpq Total loading time: 0 Render date: 2026-07-17T16:00:53.644Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic entropy of transformed random walks

Published online by Cambridge University Press:  28 January 2016

BEHRANG FORGHANI
BEHRANG FORGHANI*
Affiliation:
Department of Mathematics, University of Ottawa, Canada email behrang.forghani@uconn.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (respectively, rate of escape) of the transformed random walks is equal to the asymptotic entropy (respectively, rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov formula from ergodic theory; its particular cases were established earlier by Kaimanovich [Differential entropy of the boundary of a random walk on a group. Uspekhi Mat. Nauk38(5(233)) (1983), 187–188] and Hartman et al [An Abramov formula for stationary spaces of discrete groups. Ergod. Th. & Dynam. Sys.34(3) (2014), 837–853].

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 5 , August 2017 , pp. 1480 - 1491
Copyright
© Cambridge University Press, 2016 

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.)

Article purchase

Temporarily unavailable

References

Abramov, L. M.. The entropy of a derived automorphism. Dokl. Akad. Nauk SSSR 128 (1959), 647–650.Google Scholar
Avez, A.. Entropie des groupes de type fini. C. R. Acad. Sci. Paris Sér. A–B 275 (1972), A1363–A1366.Google Scholar
Avez, A.. Théorème de Choquet–Deny pour les groupes à croissance non exponentielle. C. R. Acad. Sci. Paris Sér. A 279 (1974), 25–28.Google Scholar
Blachère, S. and Brofferio, S.. Internal diffusion limited aggregation on discrete groups having exponential growth. Probab. Theory Related Fields 137(3–4) (2007), 323–343.CrossRefGoogle Scholar
Blachère, S., Haïssinsky, P. and Mathieu, P.. Asymptotic entropy and Green speed for random walks on countable groups. Ann. Probab. 36(3) (2008), 1134–1152.CrossRefGoogle Scholar
Derriennic, Y.. Quelques applications du théorème ergodique sous-additif. Conference on Random Walks (Kleebach, 1979) (Astérisque, 74) . Société Mathématique de France, Paris, 1980, pp. 183–201 (in French).Google Scholar
Doob, J. L.. Stochastic Processes (Wiley Classics Library) . John Wiley, New York, 1990, reprint of the 1953 original, a Wiley–Interscience publication.Google Scholar
Forghani, B. and Kaimanovich, V. A.. Boundary preserving transformations of random walks, in preparation.Google Scholar
Forghani, B.. Transformed random walks. PhD Thesis, University of Ottawa, Canada, 2015.Google Scholar
Furstenberg, H.. Random Walks and Discrete Subgroups of Lie Groups (Advances in Probability and Related Topics, 1) . Marcel Dekker, New York, 1971, pp. 1–63.Google Scholar
Guivarc’h, Y.. Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire. Conference on Random Walks (Kleebach, 1979) (Astérisque, 74) . Société Mathématique de France, Paris, 1980, pp. 47–98 (in French).Google Scholar
Hartman, Y., Lima, Y. and Tamuz, O.. An Abramov formula for stationary spaces of discrete groups. Ergod. Th. & Dynam. Sys. 34(3) (2014), 837–853.CrossRefGoogle Scholar
Kac, M.. On the notion of recurrence in discrete stochastic processes. Bull. Amer. Math. Soc. 53 (1947), 1002–1010.CrossRefGoogle Scholar
Kaimanovich, V. A.. Differential entropy of the boundary of a random walk on a group. Uspekhi Mat. Nauk 38(5(233)) (1983), 187–188.Google Scholar
Kaimanovich, V. A.. Poisson boundaries of random walks on discrete solvable groups. Probability Measures on Groups, X (Oberwolfach, 1990). Plenum, New York, 1991, pp. 205–238.CrossRefGoogle Scholar
Kaimanovich, V. A.. Discretization of bounded harmonic functions on Riemannian manifolds and entropy. Potential Theory (Nagoya, 1990). de Gruyter, Berlin, 1992, pp. 213–223.CrossRefGoogle Scholar
Kaimanovich, V. A.. The Poisson formula for groups with hyperbolic properties. Ann. of Math. (2) 152(3) (2000), 659–692.CrossRefGoogle Scholar
Karlsson, A. and Ledrappier, F.. Linear drift and Poisson boundary for random walks. Pure Appl. Math. Q. 3(4) (2007), 1027–1036 (Special Issue: In honor of Grigory Margulis. Part 1).CrossRefGoogle Scholar
Kolmogorov, A. N.. A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl. Akad. Nauk SSSR (N.S.) 119 (1958), 861–864.Google Scholar
Kaimanovich, V. A. and Vershik, A. M.. Random walks on discrete groups: boundary and entropy. Ann. Probab. 11(3) (1983), 457–490.CrossRefGoogle Scholar
Rohlin, V. A.. On the fundamental ideas of measure theory. Trans. Amer. Math. Soc. 1952(71) (1952), 1–54.Google Scholar
Shannon, C. E.. A mathematical theory of communication. Bell Syst. Tech. J. 27 (1948), 379–423, 623–656.CrossRefGoogle Scholar
Sinai, Ja.. On the concept of entropy for a dynamic system. Dokl. Akad. Nauk SSSR 124 (1959), 768–771.Google Scholar
Varopoulos, N. Th.. Long range estimates for Markov chains. Bull. Sci. Math. (2) 109(3) (1985), 225–252.Google Scholar
Willis, G. A.. Probability measures on groups and some related ideals in group algebras. J. Funct. Anal. 92(1) (1990), 202–263.CrossRefGoogle Scholar