On the surviving probability of an annihilating branching process and application to a nonlinear voter model

Smail Alili and Irina Ignatiouk-Robert

Stochastic Processes and their Applications, 2001, vol. 93, issue 2, 301-316

Abstract: We study a discrete time interacting particle system which can be considered as an annihilating branching process on where at each time every particle either performs a jump as a nearest neighbor random walk, or splits (with probability [var epsilon]) into two particles which will occupy the nearest neighbor sites. Furthermore, if two particles come to the same site, then they are removed from the system. We show that if the branching probability [var epsilon]>0 is small enough, and the number of particles at initial time is finite, then the surviving probability, i.e. the probability p(t) that there is at least one particle at time t decays to zero exponentially fast. This result is applied to a nonlinear discrete time voter model (in a random and nonrandom environment) obtained as a small perturbation with parameter [var epsilon] of the classical voter model. For this class of models, we show that if [var epsilon]>0 is small enough, then the process converges to a unique invariant probability measure independently on the initial distribution. It is known that the classical one-dimensional voter model (in a random environment as well as without environment) is not ergodic, that is there exist at least two extremal invariant probability measures. Our results prove therefore the phase transition in [var epsilon]=0.

Keywords: Voter; model; Duality; Annihilating; branching; processes; Lyapunov; functions; Supermartingale (search for similar items in EconPapers)
Date: 2001
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304-4149(00)00101-0
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:93:y:2001:i:2:p:301-316

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().