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Nonhomogeneous random walks systems on ℤ

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Elcio Lebensztayn ,
Fábio Prates Machado  and
Mauricio Zuluaga Martinez
Elcio Lebensztayn*
Affiliation:
University of São Paulo
Fábio Prates Machado*
Affiliation:
University of São Paulo
Mauricio Zuluaga Martinez*
Affiliation:
Federal University of Pernambuco
*
∗Postal address: Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, CEP 05508-090, São Paulo, Brazil.
∗Postal address: Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, CEP 05508-090, São Paulo, Brazil.
∗∗∗∗Postal address: Department of Statistics, Federal University of Pernambuco, Cidade Universitária, CEP 50740-540, Recife, PE, Brazil. Email address: zuluaga@ime.usp.br
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Abstract

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We consider a random walks system on ℤ in which each active particle performs a nearest-neighbor random walk and activates all inactive particles it encounters. The movement of an active particle stops when it reaches a certain number of jumps without activating any particle. We prove that if the process relies on efficient particles (i.e. those particles with a small probability of jumping to the left) being placed strategically on ℤ, then it might survive, having active particles at any time with positive probability. On the other hand, we may construct a process that dies out eventually almost surely, even if it relies on efficient particles. That is, we discuss what happens if particles are initially placed very far away from each other or if their probability of jumping to the right tends to 1 but not fast enough.

Keywords

Random walkepidemic modelfrog model

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 562 - 571
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Bertacchi, D. and Zucca, F. (2008). Critical behaviors and critical values of branching random walks on multigraphs. J. Appl. Prob. 45, 481–497.Google Scholar
[2] Bremaud, P. (1999). {Markov chains. Gibbs fields, Monte Carlo Simulation, and Queues} (Texts Appl. Math. 31). Springer, New York.CrossRefGoogle Scholar
[3] Gantert, N. and Schmidt, P. (2009). Recurrence for the frog model with drift on {Z}. Markov Process. Relat. Fields 15, 51–58.Google Scholar
[4] Kurtz, T. G., Lebensztayn, E., Leichsenring, A. R. and Machado, F. P. (2008). Limit theorems for an epidemic model on the complete graph. ALEA 4, 45–55.Google Scholar
[5] Lebensztayn, E., Machado, F. P. and Martinez, M. Z. (2008). Random walks systems with killing on {Z}. Stochastics 80, 451–457.CrossRefGoogle Scholar