树上$λ$ -调和函数与多调和函数的边界表示
Boundary representations of $λ$-harmonic and polyharmonic functions on trees
摘要：
在可数树$T$上，允许顶点具有无限度，考虑任意随机不可约最近邻转移算子$P$ .我们给出了$P$的一般本征函数的边界积分表示，本征值为$λ\in \mathbb{C}$。这是可能的，只要$λ$是在$P$的预解集中，作为一个自伴算子在一个合适的$\ell^2$ -空间上，并且预解的对角元素（“格林函数“）在$λ$不为零。我们发现，当$P$是不变的传递（不一定是不动点自由）组的作用下，后者的条件下成立的所有$λ\ne 0$的预解集。这些结果推广和完善了Cartier、Figà-Talamanca和Steger以及Woess的先前结果。对于这些特征值，我们还提供了一个积分表示的$λ$ -多调和函数的任何顺序$n$，即功能$f: T \to \mathbb{C}$，其中$(λ\cdot I - P)^n f=0$。这是科恩等人工作的一个深远的延伸，他为齐次树上的简单随机游动和特征值$λ=1$提供了这样一种表示。最后，我们解释了（更简单）类似的结果“向前只”的过渡运营商，有时也称为树上的鞅。
英文摘要：
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 $λ\in \mathbb{C}$. This is possible whenever $λ$ 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 $λ$. We show that when $P$ is invariant under a transitive (not necessarily fixed-point-free) group action, the latter condition holds for all $λ\ne 0$ 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 $(λ\cdot I - P)^n f=0$. 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 $λ=1$. Finally, we explain the (much simpler) analogous results for "forward only" transition operators, sometimes also called martingales on trees.
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