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VOL. 16 | 1987 The minimal Martin boundary of a Cartesian product of trees
Massimo A. Picardello, Peter Sjögren

Editor(s) Michael Cowling, Christopher Meaney, William Moran

Abstract

On a tree, the Martin boundary for positive eigenfunctions of the "Laplacian" or other suitable difference operators is known to coincide with the natural boundary of the tree. In this survey, operators on a finite product of trees are considered. Old and new results are described. In the case when all the trees are homogeneous, we let the operator be a positive linear combination of the Laplacians in the factor trees. If at least one of the trees is not $\mathbb{Z}$, the corresponding Martin boundary is nontrivial for all sufficiently large eigenvalues, and is given as the product of the natural boundaries of the trees times a hyper surface which depends on the eigemalue. The situation is similar to that of a polydisc. There is a pointwise convergence theorem at the boundary. For $\mathbb{Z}^n$ however, the boundary is a one-point set. To get a nontrivial boundary here, one can consider instead an operator with drift.

Information

Published: 1 January 1987
First available in Project Euclid: 18 November 2014

zbMATH: 0664.60074
MathSciNet: MR954000

Rights: Copyright © 1987, Centre for Mathematical Analysis, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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