Asymptotic behavior of a stochastic growth process associated with a system of interacting branching random walks

作者：Ramirez AF^*; Sidoravicius V

来源：Comptes Rendus Mathematique, 2002, 335(10): 821-826.

DOI：10.1016/S1631-073X(02)02568-2

We study a continuous time growth process on Z(d) (d greater than or equal to 1) associated to the following interacting particle system: initially there is only one simple symmetric continuous time random walk of total jump rate one located at the origin; then, whenever a random walk visits a site still unvisited by any other random walk, it creates a new independent random walk starting from that site. Let us call P-d the law of such a process and S-d(0)(t) the set of sites, visited by all walks by time t. We prove that there exists a bounded, non-empty, convex set C-d subset of R-d, such that for every epsilon > 0, P-d-a.s. eventually in t, the set s(d)(0)(t) is within an epsilon neighborhood of the set [C(d)t], where for A subset of R-d we define [A] := A boolean AND Z(d). Moreover, for d large enough, the set C-d is not a ball under the Euclidean norm. We also show that the empirical density of particles within S-d(0)(t) converges weakly to a product Poisson measure of parameter one.

• 出版日期2002-11-15