Multiple boundary representations of $\lambda$-harmonic functions on trees

We consider a countable tree $T$, possibly having vertices with infinite degree, and an arbitrary stochastic nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with eigenvalue $\lambda \in \mathbb{C}$, under the condition that the oriented edges can be equipped with complex-valued weights satisfying three natural axioms. These axioms guarantee that one can construct a $\lambda$-Poisson kernel. The boundary integral is with respect to distributions, that is, elements in the dual of the space of locally constant functions. Distributions are interpreted as finitely additive complex measures. In general, they do not extend to $\sigma$-additive measures, but in any case, a local summability condition over the neighbours of each vertex is required. 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 kernel") do not vanish at $\lambda$, one can use the ordinary edge weights corresponding to the Green function, and one gets the ordinary $\lambda$-Martin kernel. We then consider the case when $P$ is invariant under a transitive group action. In this situation, we study the phenomenon that in addition to the $\lambda$-Martin kernel, there may be further choices for the edge weights which give rise to another $\lambda$-Poisson kernel with associated integral representations. In particular, we compare the resulting distributions on the boundary. The material presented here is closely related with the one of our "sister" paper arXiv:1802.01976