Abstract
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 λ ∈ C. This is possible whenever λ is in the resolvent set of P as a self-adjoint operator on a suitable ℓ2-space and the diagonal elements of the resolvent (“Green function”) do not vanish at λ. We show that when P is invariant under a transitive (not necessarily fixed-point-free) group action, the latter condition holds for all λ≠ 0 in the resolvent set. These results extend and complete previous results by Cartier, by Figà-Talamanca and Steger, and by Woess. For those eigenvalues, we also provide an integral representation of λ-polyharmonic functions of any order n, that is, functions \(f: T \to \mathbb {C}\) for which (λ ⋅ I − P)nf = 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 λ = 1. Finally, we explain the (much simpler) analogous results for “forward only” transition operators, sometimes also called martingales on trees.
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Acknowledgements
The authors are grateful to Thomas Hirschler for pointing out that a sign had been mistaken in the derivation formulas of Section ??.
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Supported by Austrian Science Fund projects FWF P24028 and W1230. The first author acknowledges support as a distinguished visiting scientist at TU Graz, and partial suppport from MIUR Excellence Department Project funds, awarded by MIUR to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
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Picardello, M.A., Woess, W. Boundary Representations of λ-Harmonic and Polyharmonic Functions on Trees. Potential Anal 51, 541–561 (2019). https://doi.org/10.1007/s11118-018-9723-5
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DOI: https://doi.org/10.1007/s11118-018-9723-5
Keywords
- Tree
- Stochastic transition operator
- λ-harmonic functions
- Polyharmonic functions
- Martin kernel
- Boundary integral