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Random walks on the affine group of local fields and of homogeneous trees

Donald I. Cartwright, Vadim A. Kaimanovich, Wolfgang Woess (1994)

Annales de l'institut Fourier

The affine group of a local field acts on the tree 𝕋 ( 𝔉 ) (the Bruhat-Tits building of GL ( 2 , 𝔉 ) ) with a fixed point in the space of ends ∂ 𝕋 ( F ) . More generally, we define the affine group Aff ( 𝔉 ) of any homogeneous tree 𝕋 as the group of all automorphisms of 𝕋 with a common fixed point in ∂ 𝕋 , and establish main asymptotic properties of random products in Aff ( 𝔉 ) : (1) law of large numbers and central limit theorem; (2) convergence to ∂ 𝕋 and solvability of the Dirichlet problem at infinity; (3) identification of the Poisson boundary...

Range of the horocyclic Radon transform on trees

Enrico Casadio Tarabusi, Joel M. Cohen, Flavia Colonna (2000)

Annales de l'institut Fourier

In this paper we study the Radon transform R on the set ℋ of horocycles of a homogeneous tree T , and describe its image on various function spaces. We show that the functions of compact support on ℋ that satisfy two explicit Radon conditions constitute the image under R of functions of finite support on T . We extend these results to spaces of functions with suitable decay on T , whose image under R satisfies corresponding decay conditions and contains distributions on ℋ that are not defined pointwise....

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