## 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 $$nd-(\nu_1+\dots+\nu_n)<d$$.

### MSC:

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

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