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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.
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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 

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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