Measure-valued discrete branching Markov processes

TL;DR

This paper develops a framework for constructing and analyzing measure-valued discrete branching Markov processes, extending superprocesses to finite configurations with potential theoretical tools for path regularity.

Contribution

It introduces a novel construction of branching Markov processes on finite configurations using superharmonic functions, advancing the theoretical understanding of measure-valued processes.

Findings

Established existence of path regularity for the processes

Extended superprocesses to finite configuration spaces

Utilized potential theory for process analysis

Abstract

We construct and study branching Markov processes on the space of finite configurations of the state space of a given standard process, controlled by a branching kernel and a killing one. In particular, we may start with a superprocess, obtaining a branching process with state space the finite configurations of positive finite measures on a topological space. A main tool in proving the path regularity of the branching process is the existence of convenient superharmonic functions having compact level sets, allowing the use of appropriate potential theoretical methods.