Quasi-processes for branching Markov chains

TL;DR

This paper extends potential theory to branching Markov chains by developing branching quasi-processes, establishing their relation to excessive measures, and constructing branching random interlacements, thus advancing probabilistic analysis of these processes.

Contribution

It introduces branching quasi-processes, links them to excessive measures, and constructs branching random interlacements, providing new tools for analyzing branching Markov chains.

Findings

Branching quasi-processes characterized by occupation measures.

Isomorphism between branching quasi-processes and excessive measures under decorability.

Construction of branching random interlacements using classical interlacement techniques.

Abstract

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.