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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[1] Albeverio, S., Karwowski, W. and Zhao, X. (1999). Asymptotics and spectral results for random walks on p-adics. Stochastic Process. Appl. 83 39-59. · Zbl 0999.60072
[2] Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press. · Zbl 0861.60003
[3] Choi, G. (1994). Criteria for recurrence and transience of semistable processes. Nagoya Math. J. 134 91-106. · Zbl 0804.60064
[4] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2000). Thick points for spatial Brownian motion: multifractal analysis of occupation measure. Ann. Probab. 28 1-35. · Zbl 1130.60311
[5] Dvoretzky, A., Erd os, P. and Kakutani, S. (1958). Points of multiplicity of plane Brownian paths. Bull. Res. Council Israel Sect. F 7F 175-180.
[6] Evans, S. (1989). Local properties of Lévy processes on a totally disconnected group. J. Theoret. Probab. 2 209-259. · Zbl 0683.60010
[7] Evans, S. (1992). Polar and nonpolar sets for a tree-indexed process. Ann. Probab. 20 579-590. · Zbl 0757.60070
[8] Fitzsimmons, P. J., Fristedt, B. E. and Shepp, L. A. (1985). The set of real numbers left uncovered by random covering intervals.Wahrsch. Verw. Gebiete 70 175-189. · Zbl 0564.60008
[9] Fitzsimmons, P. J. and Salisbury, T. S. (1989). Capacity and energy for multiparameter Markovprocesses. Ann. Inst. H. Poincaré 25 325-350. · Zbl 0689.60071
[10] Fukushima, M., \?Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin. · Zbl 0838.31001
[11] Hawkes, J. (1979). Potential theory of Lévy processes. Proc. London Math. Soc. 38 335-352. · Zbl 0401.60069
[12] Kahane, J.-P. (1985). Some Random Series of Functions, 2nd ed. Cambridge Univ. Press. · Zbl 0571.60002
[13] Khoshnevisan, D., Peres, Y. and Xiao, Y. (2000). Limsup random fractals. Electron. J. Probab. 5. · Zbl 0949.60025
[14] Le Gall, J.-F. (1987). Le comportement du mouvement brownien entre les deux instants o u il passe par un point double. J. Funct. Anal. 71 246-262. · Zbl 0624.60090
[15] Le Gall, J.-F. (1987). Temps locaux d’intersection et points multiples des processus de Lévy. Séminaire de Probabilités XXI. Lecture Notes in Math. 1247 341-374. Springer, Berlin. · Zbl 0621.60077
[16] Le Gall, J.-F. (1993). A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Related Fields 95 25-46. · Zbl 0794.60076
[17] Lyons, R. (1992). Random walks, capacity and percolation on trees. Ann. Probab. 20 2043-2088. · Zbl 0766.60091
[18] Marchal, P. (1999). Th ese de Doctorat de l’Université Pierre et Marie Curie.
[19] Pemantle, R. and Peres, Y. (1995). Galton-Watson with the same mean have the same polar sets. Ann. Probab. 23 1102-1124. · Zbl 0833.60085
[20] Peres, Y. (1996). Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys. 177 417-434. · Zbl 0851.60080
[21] Taylor, S. J. (1966). Multiple points for the sample paths of the symmetric stable process.Wahrsch. Verw. Gebiete 5 247-264. · Zbl 0146.37905
[22] CNRS, DMA, École Normale Supérieure 45 rue d’Ulm 75230 Paris cedex 05 France E-mail: marchal@dmi.ens.fr
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