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March 2014 Local and global survival for nonhomogeneous random walk systems on Z
Daniela Bertacchi, Fábio Prates Machado, Fabio Zucca
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Adv. in Appl. Probab. 46(1): 256-278 (March 2014). DOI: 10.1239/aap/1396360113

Abstract

We study an interacting random walk system on Z where at time 0 there is an active particle at 0 and one inactive particle on each site n ≥ 1. Particles become active when hit by another active particle. Once activated, the particle starting at n performs an asymmetric, translation invariant, nearest neighbor random walk with left-jump probability ln. We give conditions for global survival, local survival, and infinite activation both in the case where all particles are immortal and in the case where particles have geometrically distributed lifespan (with parameter depending on the starting location of the particle). More precisely, once activated, the particle at n survives at each step with probability pn ∈ [0, 1]. In particular, in the immortal case, we prove a 0-1 law for the probability of local survival when all particles drift to the right. Besides that, we give sufficient conditions for local survival or local extinction when all particles drift to the left. In the mortal case, we provide sufficient conditions for global survival, local survival, and local extinction (which apply to the immortal case with mixed drifts as well). Analysis of explicit examples is provided: we describe completely the phase diagram in the cases ½ - ln ~ ± 1 / nα, pn = 1 and ½ - ln ~ ± 1 / nα, 1 - pn ~ 1 / nβ (where α, β > 0).

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Daniela Bertacchi. Fábio Prates Machado. Fabio Zucca. "Local and global survival for nonhomogeneous random walk systems on Z." Adv. in Appl. Probab. 46 (1) 256 - 278, March 2014. https://doi.org/10.1239/aap/1396360113

Information

Published: March 2014
First available in Project Euclid: 1 April 2014

zbMATH: 1302.60131
MathSciNet: MR3189058
Digital Object Identifier: 10.1239/aap/1396360113

Subjects:
Primary: 60G50, 60K35

Rights: Copyright © 2014 Applied Probability Trust

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Vol.46 • No. 1 • March 2014
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