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Heat diffusion on homogeneous trees (Note on a paper by G. Medolla and A. G. Setti)

Wolfgang Woess — 2001

Bollettino dell'Unione Matematica Italiana

Medolla e Setti [6] studiano l'andamento della diffusione del calore generata dal Laplaciano discreto su un albero omogeneo e dimostrano che il calore è asintoticamente concentrato in «anelli» che viaggiano verso l'infinito a velocità lineare e la cui larghezza divisa per t tende all'infinito, dove t è il tempo. Qui si spiega come un risultato più preciso si ottiene come corollario della legge dei grandi numeri e del teorema del limite centrale per la passeggiata aleatoria sull'albero. Inoltre,...

Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity

Marc PeignéWolfgang Woess — 2011

Colloquium Mathematicae

Consider a proper metric space and a sequence ( F ) n 0 of i.i.d. random continuous mappings → . It induces the stochastic dynamical system (SDS) X x = F . . . F ( x ) starting at x ∈ . In this and the subsequent paper, we study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process. In the present first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic...

Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings

Marc PeignéWolfgang Woess — 2011

Colloquium Mathematicae

In this continuation of the preceding paper (Part I), we consider a sequence ( F ) n 0 of i.i.d. random Lipschitz mappings → , where is a proper metric space. We investigate existence and uniqueness of invariant measures, as well as recurrence and ergodicity of the induced stochastic dynamical system (SDS) X x = F . . . F ( x ) starting at x ∈ . The main results concern the case when the associated Lipschitz constants are log-centered. Principal tools are local contractivity, as considered in detail in Part I, the Chacon-Ornstein...

Transition operators on co-compact G-spaces.

Laurent Saloff-CosteWolfgang Woess — 2006

Revista Matemática Iberoamericana

We develop methods for studying transition operators on metric spaces that are invariant under a co-compact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce reduced transition operators on the compact factor space whose norms and spectral radii are upper bounds for the Lp-norms and spectral radii of the original operator. If the group is amenable then the spectral radii of the original and...

Horocyclic products of trees

Laurent BartholdiMarkus NeuhauserWolfgang Woess — 2008

Journal of the European Mathematical Society

Let T 1 , , T d be homogeneous trees with degrees q 1 + 1 , , q d + 1 3 , respectively. For each tree, let 𝔥 : T j be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of T 1 , , T d is the graph 𝖣𝖫 ( q 1 , , q d ) consisting of all d -tuples x 1 x d T 1 × × T d with 𝔥 ( x 1 ) + + 𝔥 ( x d ) = 0 , equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If d = 2 and q 1 = q 2 = q then we obtain a Cayley graph of the...

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

Donald I. CartwrightVadim A. KaimanovichWolfgang 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...

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