## Electronic Journal of Probability

### Competing Particle Systems Evolving by I.I.D. Increments

Mykhaylo Shkolnikov

#### Abstract

We consider competing particle systems in $\mathbb{R}^d$, i.e. random locally finite upper bounded configurations of points in $\mathbb{R}^d$ evolving in discrete time steps. In each step i.i.d. increments are added to the particles independently of the initial configuration and the previous steps. Ruzmaikina and Aizenman characterized quasi-stationary measures of such an evolution, i.e. point processes for which the joint distribution of the gaps between the particles is invariant under the evolution, in case $d=1$ and restricting to increments having a density and an everywhere finite moment generating function. We prove corresponding versions of their theorem in dimension $d=1$ for heavy-tailed increments in the domain of attraction of a stable law and in dimension $d\geq 1$ for lattice type increments with an everywhere finite moment generating function. In all cases we only assume that under the initial configuration no two particles are located at the same point. In addition, we analyze the attractivity of quasi-stationary Poisson point processes in the space of all Poisson point processes with almost surely infinite, locally finite and upper bounded configurations.

#### Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 27, 728-751.

Dates
Accepted: 11 March 2009
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819488

Digital Object Identifier
doi:10.1214/EJP.v14-635

Mathematical Reviews number (MathSciNet)
MR2486819

Zentralblatt MATH identifier
1190.60039

Rights

#### Citation

Shkolnikov, Mykhaylo. Competing Particle Systems Evolving by I.I.D. Increments. Electron. J. Probab. 14 (2009), paper no. 27, 728--751. doi:10.1214/EJP.v14-635. https://projecteuclid.org/euclid.ejp/1464819488

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