×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.