Stable processes on the boundary of a regular tree. (English) Zbl 1016.60056

Let \(C_{n_0}=0,1,\dots,n_0-1= Z/n_0Z\), \(G=C_{n_0}^N\), \(G_n=((a_1,\dots,a_k,\dots)\in G: a_1=a_2=\dots=a_n=0)\). The author defines a Lévy measure on \(G\) as \(\Pi=\sum_{n=0}^\infty m^n\mu_n\) where \(m>1\) is some fixed real, \(\mu_n\) is a uniform measure on \(G_n\) with total mass 1. A stable Poisson point process \(X\) is considered on \(R_{+}\times G\) with intensity measure \(d\to\times \Pi\). Note, that \(G\) can be considered as a boundary of an \(n_0\)-regular tree. \(X\) is compared with Bernoulli percolation and a branching random walk. In the Gromov metrics with parameter \(\beta\), \(X\) has a stability index \(\nu=\log m/\log\beta\). Various results are obtained on the intersection properties and multiple points of \(X\). E.g. if \(X_1, \dots, X_n\) are independent stable processes with respective indices \(\nu_1, \dots, \nu_n<\log n_0/\log\beta=d\), then \(\text{Pr}([X_1]\cap\dots\cap[X_n])>0\) iff \(nd-(\nu_1+\dots+\nu_n)<d\).


60G52 Stable stochastic processes
60G51 Processes with independent increments; Lévy processes
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