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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)
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

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