The Annals of Probability

Markov Chains Indexed by Trees

Itai Benjamini and Yuval Peres

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Abstract

We study a variant of branching Markov chains in which the branching is governed by a fixed deterministic tree $T$ rather than a Galton-Watson process. Sample path properties of these chains are determined by an interplay of the tree structure and the transition probabilities. For instance, there exists an infinite path in $T$ with a bounded trajectory iff the Hausdorff dimension of $T$ is greater than $\log(1/\rho)$ where $\rho$ is the spectral radius of the transition matrix.

Article information

Source
Ann. Probab. Volume 22, Number 1 (1994), 219-243.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176988857

Digital Object Identifier
doi:10.1214/aop/1176988857

Mathematical Reviews number (MathSciNet)
MR1258875

Zentralblatt MATH identifier
0793.60080

JSTOR
links.jstor.org

Subjects
Primary: 60J15
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Trees Markov chains branching random walks recurrence Hausdorff dimension packing dimension

Citation

Benjamini, Itai; Peres, Yuval. Markov Chains Indexed by Trees. Ann. Probab. 22 (1994), no. 1, 219--243. doi:10.1214/aop/1176988857. http://projecteuclid.org/euclid.aop/1176988857.


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