From Brownian motion with a local time drift to Feller's branching diffusion with logistic growth

TL;DR

This paper provides a new stochastic analysis proof for a Ray-Knight representation of Feller's branching diffusion with logistic growth, connecting local times of reflected Brownian motion with the process.

Contribution

It introduces a novel proof method based solely on stochastic analysis, differing from previous approximation-based approaches.

Findings

New proof of Ray-Knight representation for Feller's branching diffusion with logistic growth.

Representation in terms of local times of reflected Brownian motion with affine linear drift.

Connections established between local time processes and branching diffusions.

Abstract

We give a new proof for a Ray-Knight representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion $H$ with a drift that is affine linear in the local time accumulated by $H$ at its current level. In \cite{LPW}, such a representation was obtained by an approximation through Harris paths that code the genealogies of particle systems. The present proof is purely in terms of stochastic analysis, and is inspired by previous work of Norris, Rogers and Williams \cite{NRW}.