The minimal Martin boundary of a Cartesian product of trees
Massimo A. Picardello, Peter Sjögren · Project Euclid (Cornell University) · 1987
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.