# The Evolution of a Spatial Stochastic Network

Abstract : The asymptotic behavior of a stochastic network represented by a birth and death processes of particles on a compact state space is analyzed. Births: Particles are created at rate $\lambda_+$ and their location is independent of the current configuration. Deaths are due to negative particles arriving at rate $\lambda_-$. The death of a particle occurs when a negative particle arrives in its neighborhood and kills it. Several killing schemes are considered. The arriving locations of positive and negative particles are assumed to have the same distribution. By using a combination of monotonicity properties and invariance relations it is shown that the configurations of particles converge in distribution for several models. The problems of uniqueness of invariant measures and of the existence of accumulation points for the limiting configurations are also investigated. It is shown for several natural models that if $\lambda_+<\lambda_-$ then the asymptotic configuration has a finite number of points with probability 1. Examples with $\lambda_+<\lambda_-$ and an infinite number of particles in the limit are also presented.
Type de document :
Pré-publication, Document de travail
2010
Domaine :

https://hal.inria.fr/inria-00511867
Contributeur : Philippe Robert <>
Soumis le : jeudi 26 août 2010 - 14:42:33
Dernière modification le : vendredi 25 mai 2018 - 12:02:03

### Identifiants

• HAL Id : inria-00511867, version 1
• ARXIV : 0908.3256

### Citation

Philippe Robert. The Evolution of a Spatial Stochastic Network. 2010. 〈inria-00511867〉

### Métriques

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