## Electronic Journal of Probability

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

Pierre Rousselin

#### 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
Accepted: 18 April 2018
First available in Project Euclid: 25 May 2018

https://projecteuclid.org/euclid.ejp/1527213727

Digital Object Identifier
doi:10.1214/18-EJP170

Mathematical Reviews number (MathSciNet)
MR3814240

Zentralblatt MATH identifier
06924658

#### 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

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