2007/05/07 by Claude Dellacherie, Dellacherie, Claude, Servet Martinez +5
Computer Science · Mathematics · #60J15 #60J45 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis #advanced mathematical theories #math.PR #msc:60J15 #msc:60J45
paper · pdf · doi:10.48550/arxiv.0705.0967
4 figures
arxiv created 2007/05/07 · openalex publication_date 2007/05/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study infinite tree and ultrametric matrices, and their action on the boundary of the tree. For each tree matrix we show the existence of a symmetric random walk associated to it and we study its Green potential. We provide a representation theorem for harmonic functions that includes simple expressions for any increasing harmonic function and the Martin kernel. In the boundary, we construct the Markov kernel whose Green function is the extension of the matrix and we simulate it by using a cascade of killing independent exponential random variables and conditionally independent uniform variables. For ultrametric matrices we supply probabilistic conditions to study its potential properties when immersed in its minimal tree matrix extension.