Electronic Journal of Probability

Invariant measures, Hausdorff dimension and dimension drop of some harmonic measures on Galton-Watson trees

Pierre Rousselin

Full-text: Open access

Abstract

We consider infinite Galton-Watson trees without leaves together with i.i.d. random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic assumptions, to build explicit invariant measures. We apply this result, together with the ergodic theory on Galton-Watson trees developed in [12], to the computation of Hausdorff dimensions of harmonic measures in two cases. The first one is the harmonic measure of the (transient) $\lambda $-biased random walk on Galton-Watson trees, for which the invariant measure and the dimension were not explicitly known. The second case is a model of random walk on a Galton-Watson trees with random lengths for which we compute the dimensions of the harmonic measure and show dimension drop phenomenon for the natural metric on the boundary and another metric that depends on the random lengths.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 46, 31 pp.

Dates
Received: 8 September 2017
Accepted: 18 April 2018
First available in Project Euclid: 25 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1527213727

Digital Object Identifier
doi:10.1214/18-EJP170

Mathematical Reviews number (MathSciNet)
MR3814240

Zentralblatt MATH identifier
06924658

Subjects
Primary: 60J50: Boundary theory 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Galton-Watson tree random walk harmonic measure Hausdorff dimension

Rights
Creative Commons Attribution 4.0 International License.

Citation

Rousselin, Pierre. Invariant measures, Hausdorff dimension and dimension drop of some harmonic measures on Galton-Watson trees. Electron. J. Probab. 23 (2018), paper no. 46, 31 pp. doi:10.1214/18-EJP170. https://projecteuclid.org/euclid.ejp/1527213727


Export citation

References

  • [1] Elie Aïdékon, Speed of the biased random walk on a Galton-Watson tree, Probab. Theory Related Fields 159 (2014), no. 3–4, 597–617.
  • [2] K. B. Athreya and P. E. Ney, Branching processes, Dover Publications, Inc., Mineola, NY, 2004, Reprint of the 1972 original [Springer, New York; MR0373040].
  • [3] Nathanaël Berestycki, Eyal Lubetzky, Yuval Peres, and Allan Sly, Random walks on the random graph, Ann. Probab. 46 (2018), no. 1, 456–490.
  • [4] Nicolas Curien and Jean-François Le Gall, The harmonic measure of balls in random trees, Ann. Probab. 45 (2017), no. 1, 147–209.
  • [5] Colleen D. Cutler, The density theorem and Hausdorff inequality for packing measure in general metric spaces, Illinois J. Math. 39 (1995), no. 4, 676–694.
  • [6] Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997.
  • [7] Peter Jagers and Olle Nerman, The growth and composition of branching populations, Adv. in Appl. Probab. 16 (1984), no. 2, 221–259.
  • [8] G. J. O. Jameson, Inequalities for gamma function ratios, Amer. Math. Monthly 120 (2013), no. 10, 936–940.
  • [9] Shen Lin, The harmonic measure of balls in critical Galton-Watson trees with infinite variance offspring distribution, Electron. J. Probab. 19 (2014), no. 98, 35.
  • [10] Shen Lin, Harmonic measure for biased random walk in a supercritical Galton-Watson tree, ArXiv e-prints (2017), arXiv:1707.01811.
  • [11] Russell Lyons, Random walks and percolation on trees, Ann. Probab. 18 (1990), no. 3, 931–958.
  • [12] Russell Lyons, Robin Pemantle, and Yuval Peres, Ergodic theory on Galton-Watson trees: speed of random walk and dimension of harmonic measure, Ergodic Theory Dynam. Systems 15 (1995), no. 3, 593–619.
  • [13] Russell Lyons, Robin Pemantle, and Yuval Peres, Biased random walks on Galton-Watson trees, Probab. Theory Related Fields 106 (1996), no. 2, 249–264.
  • [14] Russell Lyons, Robin Pemantle, and Yuval Peres, Unsolved problems concerning random walks on trees, Classical and modern branching processes (Minneapolis, MN, 1994), IMA Vol. Math. Appl., vol. 84, Springer, New York, 1997, pp. 223–237.
  • [15] Russell Lyons and Yuval Peres, Probability on trees and networks, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 42, Cambridge University Press, New York, 2016.
  • [16] N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. (3) 51 (1985), no. 2, 369–384.
  • [17] J. Neveu, Arbres et processus de Galton-Watson, Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 2, 199–207.
  • [18] Xavier Saint Raymond and Claude Tricot, Packing regularity of sets in $n$-space, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 1, 133–145.
  • [19] Lai Sang Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynamical Systems 2 (1982), no. 1, 109–124.