Markov Chains Indexed by Trees
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.
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