Boundary behaviour of $$\lambda $$-polyharmonic functions on regular trees

Abstract

AbstractThis paper studies the boundary behaviour of $$\lambda $$ λ -polyharmonic functions for the simple random walk operator on a regular tree, where $$\lambda $$ λ is complex and $$|\lambda |> \rho $$ | λ | > ρ , the $$\ell ^2$$ ℓ 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.