A path-valued Markov process indexed by the ancestral mass

TL;DR

This paper introduces a path-valued Markov process derived from a family of Feller branching diffusions with nonlinear drift, identifying its generator and analyzing its path properties through couplings and stochastic differential equations.

Contribution

It develops a novel coupling approach to view Feller branching diffusions as a path-valued Markov process and derives its SDE and infinitesimal generator.

Findings

Identified the SDE driven by a random point measure on excursion space.

Established the Markov property of the path-valued process.

Analyzed path properties using couplings with classical Feller diffusions.

Abstract

A family of Feller branching diffusions $Z^x$, $x \ge 0$, with nonlinear drift and initial value $x$ can, with a suitable coupling over the {\em ancestral masses} $x$, be viewed as a path-valued process indexed by $x$. For a coupling due to Dawson and Li, which in case of a linear drift describes the corresponding Feller branching diffusion, and in our case makes the path-valued process Markovian, we find an SDE solved by $Z$, which is driven by a random point measure on excursion space. In this way we are able to identify the infinitesimal generator of the path-valued process. We also establish path properties of $x\mapsto Z^x$ using various couplings of $Z$ with classical Feller branching diffusions.