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

Abstract

This 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.

Discrete harmonic functions for non-symmetric Laplace operators in the quarter plane

“…Almansi in [1] proved that in the continuous context, polyharmonic functions can be decomposed into signed harmonic functions. The discrete analog of this result was achieved in [12,36,8]. Further in [8], Chapon, Fusy and Raschel unveiled some connections between discrete polyharmonic functions and complete asymptotic expansions in walk enumeration problems.…”

Boundary Representations of λ-Harmonic and Polyharmonic Functions on Trees

“…So we can assume that ν is absolutely continuous with respect to the normalised Lebesgue measure m, with g = dν/dm ∈ L 1 (∂D, m). The rest of the proof is the continuous analogue of [21, Theorems 1 and 3] and [30,Theorem 4.6]. In brief, we can find a sequence (g k ) in C(∂D) such that g − g k < 1/2 k , and by propositions 6.2 and 6.3,…”

“…As a consequence, we have the following convergence theorem (in the discrete setting of trees, see [21, Theorem 1, Theorem 3] for λ-harmonic functions, and [30,Theorem 4.6] for regular trees and λ-polyharmonic functions). Definition 6.4.…”

“…(In [28], this is generalised and simplified.) Further motivation for the present work came from Sava-Huss and Woess [30], who studied the boundary behaviour of polyharmonic functions on regular trees. There are many profound analogies between the hyperbolic disk and regular trees.…”