Published in Potential Analysis
Let T be a homogeneous tree of homogeneity q+1. Let Δ denote the boundary of T, consisting of all infinite geodesics b=[b0,b1,b2,] beginning at the root, 0. For each bεΔ, τ≥1, and a≥0 we define the approach region Ωτ,a(b) to be the set of all vertices t such that, for some j, t is a descendant of bj and the geodesic distance of t to bj is at most (...
Published in Journal of Statistical Physics
We study the frog model on homogeneous trees, a discrete time system of simple symmetric random walks whose description is as follows. There are active and inactive particles living on the vertices. Each active particle performs a simple symmetric random walk having a geometrically distributed random lifetime with parameter (1 − p). When an active ...
Published in Integral Equations and Operator Theory
We consider stationary multiscale systems as defined by Basseville, Benveniste, Nikoukhah and Willsky. We show that there are deep analogies with the discrete time non stationary setting as developed by the first author, Dewilde and Dym. Following these analogies we define a point evaluation with values in a C*–algebra and the corresponding “Hardy ...
We solve explicitly the shifted wave equation on Damek--Ricci spaces, using Asgeirsson's theorem and the inverse dual Abel transform. As an application, we investigate Huygens' principle. A similar analysis is carried out in the discrete setting of homogeneous trees.
Let T be a homogeneous tree and L the Laplace operator on T. We consider the semilinear Schrodinger equation associated to L with a power-like nonlinearity F of degree d. We first obtain dispersive estimates and Strichartz estimates with no admissibility conditions. We next deduce global well-posedness for small L2 data with no gauge invariance ass...
We describe the spectral theory of the adjacency operator of a graph which is isomorphic to homogeneous trees at infinity. Using some combinatorics, we reduce the problem to a scattering problem for a finite rank perturbation of the adjacency operator on an homogeneous tree. We developp this scattering theory using the classical recipes for Schrödi...
Published in Functional Analysis and Its Applications
We describe the full exit boundary of random walks on homogeneous trees, in particular, on free groups. This model exhibits a phase transition; namely, the family of Markov measures under study loses ergodicity as a parameter of the random walk changes. The problem under consideration is a special case of the problem of describing the invariant (ce...
Published in Bulletin of Mathematical Biology
We consider an epidemic model of SIR type set on a homogeneous tree and investigate the spreading properties of the epidemic as a function of the degree of the tree, the intrinsic basic reproduction number and the strength of the interactions between the populations of infected individuals at each node. When the degree is one, the homogeneous tree ...
Published in Journal of Physics A: Mathematical and Theoretical
We study a discrete susceptible–infected–recovered (SIR) model for the spread of infectious disease on a homogeneous tree and the limit behavior of the model in the case when the tree vertex degree tends to infinity. We obtain the distribution of the time it takes for a susceptible vertex to get infected in terms of a solution of a non-linear integ...
