The Annals of Probability

Stable Processes on the Boundary of a Regular Tree

Phillippe Marchal

Full-text: Open access


We define a class of processes on the boundary of a regular tree that can be viewed as “stable” Lévy processes on $(\mathbb{Z}/n_0\mathbb{Z})^\mathbb{N}$. We show that the range of these processes can be compared with a Bernoulli percolation as in Peres which easily leads to various results on the intersection properties. We develop an alternative approach based on the comparison with a branching random walk. By this method we establish the existence of points of in finite multiplicity when the index of the process equals the dimension of the state space, as for planar Brownian motion.

Article information

Ann. Probab., Volume 29, Number 4 (2001), 1591-1611.

First available in Project Euclid: 5 March 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Marchal, Phillippe. Stable Processes on the Boundary of a Regular Tree. Ann. Probab. 29 (2001), no. 4, 1591--1611. doi:10.1214/aop/1015345763.

Export citation


  • [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.
  • [2] Bertoin, J. (1996). L´evy Processes. Cambridge Univ. Press.
  • [3] Choi, G. (1994). Criteria for recurrence and transience of semistable processes. Nagoya Math. J. 134 91-106.
  • [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.
  • [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 L´evy processes on a totally disconnected group. J. Theoret. Probab. 2 209-259.
  • [7] Evans, S. (1992). Polar and nonpolar sets for a tree-indexed process. Ann. Probab. 20 579-590.
  • [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.
  • [9] Fitzsimmons, P. J. and Salisbury, T. S. (1989). Capacity and energy for multiparameter Markovprocesses. Ann. Inst. H. Poincar´e 25 325-350.
  • [10] Fukushima, M., ¯Oshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin.
  • [11] Hawkes, J. (1979). Potential theory of L´evy processes. Proc. London Math. Soc. 38 335-352.
  • [12] Kahane, J.-P. (1985). Some Random Series of Functions, 2nd ed. Cambridge Univ. Press.
  • [13] Khoshnevisan, D., Peres, Y. and Xiao, Y. (2000). Limsup random fractals. Electron. J. Probab. 5.
  • [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.
  • [15] Le Gall, J.-F. (1987). Temps locaux d'intersection et points multiples des processus de L´evy. S´eminaire de Probabilit´es XXI. Lecture Notes in Math. 1247 341-374. Springer, Berlin.
  • [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.
  • [17] Lyons, R. (1992). Random walks, capacity and percolation on trees. Ann. Probab. 20 2043-2088.
  • [18] Marchal, P. (1999). Th ese de Doctorat de l'Universit´e 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.
  • [20] Peres, Y. (1996). Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys. 177 417-434.
  • [21] Taylor, S. J. (1966). Multiple points for the sample paths of the symmetric stable process.Wahrsch. Verw. Gebiete 5 247-264.
  • CNRS, DMA, ´Ecole Normale Sup´erieure 45 rue d'Ulm 75230 Paris cedex 05 France E-mail: