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Competing particle systems evolving by I.I.D. Increments. (English) Zbl 1190.60039
Summary: 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 \(d>1\) 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.

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G70 Extreme value theory; extremal stochastic processes
60K40 Other physical applications of random processes
82C22 Interacting particle systems in time-dependent statistical mechanics