On a countable tree T, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator P. We provide a boundary integral representation for general eigenfu...
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On a countable tree T, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator P. We provide a boundary integral representation for general eigenfunctions of P with eigenvalue lambda is an element of C. This is possible whenever lambda is in the resolvent set of P as a self-adjoint operator on a suitable l(2)-space and the diagonal elements of the resolvent ("Green function") do not vanish at lambda. We show that when P is invariant under a transitive (not necessarily fixed-point-free) group action, the latter condition holds for all lambda not equal 0 in the resolvent set. These results extend and complete previous results by Cartier, by Figa-Talamanca and Steger, and by Woess. For those eigenvalues, we also provide an integral representation of lambda-polyharmonicfunctions of any order n, that is, functions f:T -> C for which (lambda I - P)(n)f = 0. This is a far-reaching extension of work of Cohen et al., who provided such a representation for the simple random walk on a homogeneous tree and eigenvalue lambda = 1. Finally, we explain the (much simpler) analogous results for "forward only" transition operators, sometimes also called martingales on trees.
The existence and nonexistence of.-harmonicfunctions in unbounded domains of Hn are investigated. We prove that if the ( n -1)/ 2 Hausdorff measure of the asymptotic boundary of a domain is zero, then there is no bou...
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The existence and nonexistence of.-harmonicfunctions in unbounded domains of Hn are investigated. We prove that if the ( n -1)/ 2 Hausdorff measure of the asymptotic boundary of a domain is zero, then there is no bounded.-harmonic function of for.. [0,.1( Hn)], where.1( Hn) = ( n -1) 2/ 4. For these domains, we have comparison principle and some maximum principle. Conversely, for any s > ( n-1)/ 2, we prove the existence of domains with asymptotic boundary of dimension s for which there are bounded.1-harmonicfunctions that decay exponentially at infinity.
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