Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-29T04:18:10.903Z Has data issue: false hasContentIssue false
  • English
  • Français

On the recurrence of simple random walks on some fractals

Published online by Cambridge University Press:  14 July 2016

Zhou Xian-Yin
Zhou Xian-Yin*
Affiliation:
Beijing Normal University
*
∗Postal address: Department of Mathematics, Beijing Normal University, Beijing 100875, China. Research supported by National Natural Science Foundation of China.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, the recurrence or transience of simple random walks on some lattice fractals is investigated. As results, we obtain that the simple random walk on the pre-Sierpinski gasket in d dimensions is recurrent for all d ≧ 2, and on the pre-Sierpinski carpet in d dimensions it is recurrent for d = 2 and transient for all d ≧ 3.

Keywords

PRE-SIERPINSKI CARPETTRANSIENCERANDOM WALKRESISTANCE
Type
Short Communications
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 454 - 459
Copyright
Copyright © Applied Probability Trust 1992 

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

[1] Barlow, M. T. and Bass, R. F. (1989) The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. H. Poincaré 25, 225–257.Google Scholar
[2] Barlow, M. T. and Perkins, E. (1988) Brownian motion on the Sierpinski gasket. Prob. Theory Rel. Fields 79, 543–623.CrossRefGoogle Scholar
[3] Chen, M. F. (1986) Jump Processes and Particle Systems. Beijing Normal University Press.Google Scholar
[4] Doyle, P. G. and Snell, J. L. (1984) Random Walks and Electrical Networks. Mathematical Association of America, Washington, DC.CrossRefGoogle Scholar
[5] Goldstein, S. (1987) Random walks and diffusion on fractals. In Lecture Notes IMA 8, ed. Kesten, H. Google Scholar
[6] Lyons, T. (1983) A simple criterion for transience of a reversible Markov chain. Ann. Prob. 11, 393–402.CrossRefGoogle Scholar
[7] Nash-Williams, C. St J. A. (1959) Random walks and electric current in networks. Proc. Camb. Phil. Soc. 55, 181–194.CrossRefGoogle Scholar
[8] Osada, H. (1990) Isoperimetric constants and estimates of heat kernels of pre-Sierpinski gaskets. Prob. Theory. Rel. Fields 86, 469–490.CrossRefGoogle Scholar
[9] Telcs, A. (1989) Random walks on graphs, electric networks and fractals. Prob. Theory Rel. Fields 82, 435–449.CrossRefGoogle Scholar