Martin and end compactifications for non-locally finite graphs
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In this article, the authors considered a connected graph, having countably infinite vertex set X, which is permitted to have vertices of infinite degree and studied the associated harmonic functions on X and, in particular, the Martin compactification.Abstract:
We consider a connected graph, having countably infinite vertex set X, which is permitted to have vertices of infinite degree. For a transient irreducible transition matrix P corresponding to a nearest neighbor random walk on X, we study the associated harmonic functions on X and, in particular, the Martin compactification. We also study the end compactification of the graph. When the graph is a tree, we show that these compactifications coincide; they are a disjoint union of X, the set of ends, and the set of improper vertices-new points associated with vertices of infinite degree. Other results proved include a solution of the Dirichlet problem in the context of the end compactification of a general graph. Applications are given to, e.g., the Cayley graph of a free group on infinitely many- generatorsread more
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Journal ArticleDOI
Random Walks on Infinite Graphs and Groups — a Survey on Selected topics
TL;DR: In this paper, the authors define the type problem for random walks on groups and the type problems for reversible Markov chains and Dirichlet inequalities, as well as the asymptotic behaviour of transition probabilities.
Journal ArticleDOI
Isotropic Markov semigroups on ultra-metric spaces
Abstract: Let (X,d) be a locally compact separable ultra-metric space. Given a reference measure \mu\ on X and a step length distribution on the non-negative reals, we construct a symmetric Markov semigroup P^t acting in L^2(X,\mu). We study the corresponding Markov process. We obtain upper and lower bounds of its transition density and its Green function, give a transience criterion, estimate its moments and describe the Markov generator and its spectrum, which is pure point. In the particular case when X is the field of p-adic numbers, our construction recovers fractional derivative and the Taibleson Laplacian (spectral multiplier), and we can also apply our theory to the study of the Vladimirov Laplacian which is closely related to the concept of p-adic Quantum Mechanics. Even in this well established setting, several of our results are new. We also elaborate the relation between our processes and Kigami's jump processes on the boundary of a tree which are induced by a random walk. In conclusion, we provide examples illustrating the interplay between the fractional derivatives and random walks.
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Graph-theoretical versus topological ends of graphs
Reinhard Diestel,Daniela Kühn +1 more
TL;DR: It is found that the topological ends of a graph are precisely the undominated of its graph-theoretical ends, and that graph theoretical ends have a simple topological description generalizing the definition of a topological end.
Book ChapterDOI
Ergodic Theory of ℤ d Actions: Boundaries of invariant Markov Operators: The identification problem
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Book
Denumerable Markov chains
TL;DR: Giffeath as discussed by the authors provides a systematic treatment of denumerable Markov chains, covering both the foundations of the subject and topics in potential theory and boundary theory, including a new chapter, "Introduction to Random Fields", written by David Giffeath.