2018/06/11 by Daniela Bertacchi, Bertacchi, Daniela, Fabio Zucca +1
Mathematics · #60J80 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1806.04101
openalex publication_date 2018/06/11 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Given a branching random walk on a set X, we study its extinction\nprobability vectors mathbf q(\⋅,A). Their components are the probability\nthat the process goes extinct in a fixed A\⊆ X, when starting from a\nvertex x\∈ X. The set of extinction probability vectors (obtained letting\nA vary among all subsets of X) is a subset of the set of the fixed points\nof the generating function of the branching random walk. In particular here we\nare interested in the cardinality of the set of extinction probability vectors.\nWe prove results which allow to understand whether the probability of\nextinction in a set A is different from the one of extinction in another set\nB. In many cases there are only two possible extinction probability vectors\nand so far, in more complicated examples, only a finite number of distinct\nextinction probability vectors had been explicitly found. Whether a branching\nrandom walk could have an infinite number of distinct extinction probability\nvectors was not known. We apply our results to construct examples of branching\nrandom walks with uncountably many distinct extinction probability vectors.\n