A Williams' Decomposition for Spatially Dependent Superprocesses

TL;DR

This paper develops a genealogy framework for spatially dependent superprocesses with non-homogeneous branching, utilizing a weighted superprocess and Girsanov theorem, and introduces a Williams' decomposition with applications to various models.

Contribution

It introduces a Williams' decomposition for superprocesses with spatial dependence, extending genealogical analysis to non-homogeneous branching mechanisms.

Findings

Derived a genealogy for non-homogeneous superprocesses.

Established Williams' decomposition with respect to the last individual.

Applied the framework to multitype Feller diffusion and superdiffusion.

Abstract

We present a genealogy for superprocesses with a non-homogeneous quadratic branching mechanism, relying on a weighted version of the superprocess and a Girsanov theorem. We then decompose this genealogy with respect to the last individual alive (William's decomposition). Letting the extinction time tend to infinity, we get the Q-process by looking at the superprocess from the root, and define another process by looking from the top. Examples including the multitype Feller diff usion and the superdiffusion are provided.