2021/07/14 by Steffen Dereich, Dereich, Steffen, Martin Maiwald +1
Mathematics · #60J50 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Point processes and geometric inequalities #Primary 60J80 #Probability (math.PR) #Secondary 60J45 #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2107.06654
openalex publication_date 2021/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Potential theory is a central tool to understand and analyse Markov processes. In this article, we develop its probabilistic counterpart for branching Markov chains. Specifically, we examine versions of quasi-processes or interlacements that incorporate branching, referred to as branching quasi-processes. These processes are characterized by their occupation measures. If a certain decorability condition is fulfilled, there's an isomorphism between the set of branching quasi-processes and the set of excessive measures, where the excessive measures correspond to the occupation measures of the branching quasi-processes. Utilizing a branching quasi-process as an intensity measure for a Poisson point process leads to the formulation of random interlacements with branching. In cases where individuals reproduce with an average rate of one or less, we detail a construction that draws on classical interlacements. Our approach significantly employs the additive structure of branching Markov chains, with the spinal representation of the branching processes serving as a crucial technical tool.