Spine decomposition for branching Markov processes and its applications

TL;DR

This paper extends the spine decomposition technique for branching Markov processes to include cases where individuals may have no offspring, broadening its applicability and demonstrating its usefulness through various applications.

Contribution

It provides a detailed construction of spine decomposition for general branching Markov processes, including cases with zero offspring, which was not previously addressed.

Findings

Extended spine decomposition to processes with possible zero offspring

Provided applications demonstrating the method's utility

Enhanced understanding of branching Markov process structures

Abstract

In the literature, the spine decomposition of branching Markov processes was constructed under the assumption that each individual has at least one child. In this paper, we give a detailed construction of the spine decomposition of general branching Markov processes allowing the possibility of no offspring when a particle dies. Then we give some applications of the spine decomposition.