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Markov Chains Indexed by Trees

# Markov Chains Indexed by Trees,10.1214/aop/1176988857,Annals of Probability,Itai Benjamini,Yuval Peres

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.
Journal: Annals of Probability - ANN PROBAB , vol. 22, no. 1994, pp. 219-243, 1994
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## Citation Context (28)

• ...The notion of a tree-indexed random walk was rst introduced and studied by Benjamini and Peres [4]...

### Benny Sudakov, et al. A randomized embedding algorithm for trees

• ...Athreya and Kang [4], Benjamini and Peres [8]), where for every x, F(1)(x, θ) and F(2)(x, θ) are i.i.d...
• ...The law of large numbers that we obtain is a continuous time version of the law of large numbers in Benjamini and Peres [8], Delmas and Marsalle [17], with possible asymmetric branching and random number of offspring...

### Vincent Bansaye, et al. Limit theorems for Markov processes indexed by continuous time Galton-...

• ...This subject has been studied in the literature (see e.g., [6, 8]) in the symmetric independent case...

### Vincent Bansaye. Proliferating parasites in dividing cells: Kimmel’s branching model re...

• ...Definition 1 [1] . Let T be an infinite tree, S be a finite state space, {Xt ,t ∈ T } be a collection...
• ...The subject of tree-indexed processes is rather young. Benjamini and Peres [1] have given...

### Huilin Huang, et al. Strong law of large numbers for Markov chains indexed by an infinite t...

• ...Now, there are three possible regimes for irreducible BMC: transient ( (x) = 0 8x), weakly recurrent (0 < (x) < 1 for some x) and strongly recurrent ( (x) = 1 8x), compare with Gantert and M¨uller [4] and Benjamini and Peres [1]...
• ...We define recurrence and transience for BMC in analogy to [1] and [4]:...
• ...In fact, due to the irreducibility, (x) > 0 and (x) = 0 hold either for all or none x 2 X. This can be shown analogously to Lemma 3.1 in [1]...

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## Citations (48)

### Inhibition of human drug metabolizing cytochrome P450 enzymes by plant isoquinoline alkaloids(Citations: 2)

Journal: Phytomedicine , vol. 18, no. 6, pp. 533-538, 2011

### A randomized embedding algorithm for trees(Citations: 2)

Journal: Combinatorica , vol. 30, no. 4, pp. 445-470, 2010

### Branching Feller diffusion for cell division with parasite infection(Citations: 2)

Published in 2010.