### The Evolution of a Spatial Stochastic Network

**Robert, P ***(INRIA Paris - Rocquencourt)*

Friday 26 March 2010, 11:00-12:00

Seminar Room 1, Newton Institute

#### Abstract

The asymptotic behavior of a stochastic network represented by a birth and death processes of particles is analyzed. Births: Particles are created at rate $\l_+$ and their location is independent of the current configuration. Deaths are due to negative particles arriving at rate $\l_-$. 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 $\l_+

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