This paper studies the boundary behaviour of \(\lambda \)-polyharmonic functions for the simple random walk operator on a regular tree, where \(\lambda \) is complex and \(|\lambda |> \rho \), the \(\ell ^2\)-spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved, and a non-tangential Fatou theorem is proved.

中文翻译：

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$$\lambda $$λ - 正则树上的多调和函数的边界行为

本文研究了正则树上简单随机游走算子的\(\lambda \)多调和函数的边界行为，其中\(\lambda \)是复数且\(|\lambda |> \rho \) ，随机游走的\(\ell ^2\)谱半径。特别是，分别通过球面进行归一化。解决了无穷远的多球函数、Dirichlet 和 Riquier 问题，并证明了非切线 Fatou 定理。