Williams decomposition for superprocesses

TL;DR

This paper generalizes Williams decomposition for superprocesses with spatially dependent branching, enabling analysis of their genealogy and mass extinction behavior, extending prior quadratic branching results to more general mechanisms.

Contribution

It extends Williams decomposition to superprocesses with general spatially dependent branching mechanisms, broadening the scope of genealogical analysis.

Findings

Normalized total mass converges to a point mass at extinction time.

Generalizes previous quadratic branching results.

Provides a new tool for analyzing superprocess genealogy.

Abstract

We decompose the genealogy of a general superprocess with spatially dependent branching mechanism with respect to the last individual alive (Williams decomposition). This is a generalization of the main result of Delmas and H\'{e}nard [Electron. J. Probab.,18,1-43,2013] where only superprocesses with spatially dependent quadratic branching mechanism were considered. As an application of the Williams decomposition, we prove that, for some superprocesses, the normalized total mass will converge to a point mass at its extinction time. This generalizes a result of Tribe [Ann. Probab.,20,286-311,1992] in the sense that our branching mechanism is more general.