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.
Highlights
Let T be a countable tree without leaves
The same holds for λ = ρ(P ) in case P is ρ-transient. For those eigenvalues one has a similar integral representation for any real eigenfunction, where the integral of the Martin kernel is taken with respect to a distribution, that is, a finitely additive signed measure which is defined on the collection of all boundary arcs
There is one restriction for the general representation, namely, that λ has to be such that the diagonal matrix elements G(x, x|λ) of the resolvent operator G(λ) = (λ · I − P )−1 do not vanish. This holds always when |λ| ≥ ρ(P ). (The use of the letter G is motivated by the usual name “Green function” for its matrix elements.) We show that the condition on λ allows us to construct the general analogue of the λ-Martin kernel
Summary
Let T be a countable tree without leaves (vertices with only one neighbour). We allow vertices with countably many neighbours. The same holds for λ = ρ(P ) in case P is ρ-transient For those eigenvalues one has a similar integral representation for any real (or complex) eigenfunction, where the integral of the Martin kernel is taken with respect to a distribution, that is, a finitely additive signed measure which is defined on the collection of all boundary arcs. For the special case of the simple random walk on a locally finite, homogeneous tree, they provide a boundary integral representation for polyharmonic functions. We provide a far-reaching generalisation: in Theorem 5.3 we explain how this can by achieved more directly for arbitrary λ-polyharmonic functions in the general setting of a nearest neighbour random walk on a countable tree T (locally finite or not), whenever λ ∈ res(P ) fulfils the above condition that G(x, x|λ) = 0 for every x ∈ T.
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