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Uniform asymptotic estimates of transition probabilities on combs

Part of: Markov processes

Published online by Cambridge University Press:  09 April 2009

Daniela Bertacchi  and
Fabio Zucca
Daniela Bertacchi
Affiliation:
Università di Milano-BicoccaDipartimento di Matematica e Applicazioni Via Bicocca degli Arcimboldi 8 20126 Milano, Italy e-mail: bertacchi@matapp.unimib.it
Fabio Zucca
Affiliation:
Politecnico di Milano, Dipartimento di Matematica, Piazza Leonardo da Vinci 32 20133 Milano, Italy e-mail: zucca@mate.polimi.it
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Abstract

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We investigate the asymptotical behaviour of the transition probabilities of the simple random walk on the 2-comb. In particular, we obtain space-time uniform asymptotical estimates which show the lack of symmetry of this walk better than local limit estimates. Our results also point out the impossibility of getting sub-Gaussian estimates involving the spectral and walk dimensions of the graph.

Keywords

uniform estimateuniform Lebesgue theoremGreen functiontransition probabilityCauchy integralcomb

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 3 , December 2003 , pp. 325 - 354
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Barlow, M. T. and Bass, R. F., ‘Random walks on graphical Sierpiński carpets’, in: Random walks and discrete potential theory (Cortona, 1997), Sympos. Math. 39 (Cambridge Univ. Press, Cambridge, 1999) pp. 26–55.Google Scholar
[2]Bender, E. A., ‘Asymptotic method in enumeration’, SIAM Rev. 4 (1974), 485–515.CrossRefGoogle Scholar
[3]Bertacchi, D. and Zucca, F., ‘Equidistribution of random walks on spheres’, J. Stat. Phys. 94 (1999), 91–111.CrossRefGoogle Scholar
[4]Cartwright, D. I., ‘Some examples of random walks on free products of discrete groups’, Ann. Mat. Pura Appl. (4) 151 (1988), 1–15.CrossRefGoogle Scholar
[5]Cartwright, D. I. and Soardi, P. M., ‘Random walks on free products, quotients and amalgams’, Nagoya Math. J. 102 (1986), 163–180.CrossRefGoogle Scholar
[6]Catwright, D. I. and Soardi, P. M., ‘A local limit theorem for random walks on the Cartesian product of disrete groups’, Boll. Un. Mat. Ital. A (7) 1 (1987), 107–115.Google Scholar
[7]Cassi, D. and Regina, S., ‘Random walks on d—dimensional comb lattices’, Modern Phys. Lett. B 6 (1992), 1397–1403.CrossRefGoogle Scholar
[8]Gerl, P., ‘Natural spanning trees of Zd are recurrent’, Discrete Math. 61 (1986), 333–336.CrossRefGoogle Scholar
[9]Grigor'yan, A. and Telcs, A., ‘Sub-Gaussian estimates of heat kernels on infinite graphs’, Duke Math. J. 109 (2001), 451–510.CrossRefGoogle Scholar
[10]Havlin, S. and Weiss, G. H., ‘Some properties of a random walk on a comb structure’, Phys. A 134 (1986), 474–482.Google Scholar
[11]Jones, O. D., ‘Transition probabilities for the simple random walk on the Sierpiński graph’, Stochastic Process. Appl. 61 (1996), 4–69.CrossRefGoogle Scholar
[12]Lalley, S. P., ‘Saddlepoint approximations and space-time Martin boundary for nearest neighbour random walk on a homogeneous tree’, J. Theoret. Probab. 4 (1991), 701–723.CrossRefGoogle Scholar
[13]Lalley, S. P., ‘Finite range random walks on free groups and homogeneous trees’, Ann. Probab. 21 (1993), 2087–2130.CrossRefGoogle Scholar
[14]Telcs, A., ‘Spectra of graphs and fractal dimensions. I’, Probab. Theory Related Fields 85 (1990), 489–497.CrossRefGoogle Scholar
[15]Telcs, A., ‘Spectra of graphs and fractal dimensions. II’, Probab. Theory Related Fields 8 (1995), 77–96.Google Scholar
[16]Telcs, A., ‘Local sub-Gaussian estimates on graphs, the strongly recurrent case’, Electron. J. Probab. 6 (2001), 33.CrossRefGoogle Scholar
[17]Woess, W., ‘Nearest neighbour random walks on free product of discrete groups’, Boll. Un. Mat. Ital. B (6) 5 (1986), 961–982.Google Scholar
[18]Woess, W., Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics 138 (Cambridge Univ. Press, Cambridge, 2000).CrossRefGoogle Scholar