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Register a new userMultiple boundary representations of $lambda$-harmonic functions on trees
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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: for this extension, a summability condition over disjoint boundary arcs is required. Whenever $lambda$ is in the resolvent of $P$ as a self-adjoint operator on a naturally associated $ell^2$-space and the diagonal elements of the resolvent (`Green function) do not vanish at $lambda$, one can use the ordinary edge weights corresponding to the Green function and obtain 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 to the contents of our `companion paper arXiv:1802.01976
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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 $lambda in mathbb{C}$. 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 $lambda e 0$ 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.
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