Boundary representations of $\lambda$-harmonic and polyharmonic functions on trees

TL;DR

This paper develops boundary integral representations for eigenfunctions and polyharmonic functions on trees with arbitrary transition operators, extending classical results to more general settings including infinite degree vertices and group-invariant operators.

Contribution

It introduces a boundary integral framework for $ ext{lambda}$-harmonic and polyharmonic functions on trees with general transition operators, broadening previous theories to include infinite degrees and group invariance.

Findings

Boundary representation for eigenfunctions with eigenvalues in the resolvent set.

Extension of integral representations to polyharmonic functions of any order.

Results apply to trees with infinite degree vertices and group-invariant transition operators.

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 $\lambda \in \mathbb{C}$. This is possible whenever $\lambda$ is in the resolvent set of $P$ as a self-adjoint operator on a suitable $\ell^2$-space and the on-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 $\lambda \ne 0$ in the resolvent set. These results extend and complete previous results by Cartier, by Fig\`a-Talamanca and Steger, and by Woess. For those eigenvalues, we also provide an integral representation of $\lambda$-polyharmonic functions of any order $n$, that is, functions $f: T \to \mathbb{C}$ for which $(\lambda \cdot I - P)^n f=0$. This is a far-reaching extension of work of Cohen et al., who provided such a representation for 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.