An invariance principle for branching diffusions in bounded domains

TL;DR

This paper proves that at criticality, the genealogical structure of branching diffusions in bounded domains converges to Aldous' Continuum Random Tree, extending understanding of phase transitions and scaling limits in such stochastic processes.

Contribution

It establishes a convergence result to the Continuum Random Tree for critical branching diffusions in bounded domains under mild assumptions, broadening the scope of known scaling limits.

Findings

Genealogical trees converge to Aldous' CRT at criticality

Results hold for general diffusions with finite variance

Applicable to a wide class of bounded domains

Abstract

We study branching diffusions in a bounded domain $D$ of $\mathbb{R}^d$ in which particles are killed upon hitting the boundary $\partial D$. It is known that any such process undergoes a phase transition when the branching rate $\beta$ exceeds a critical value: a multiple of the first eigenvalue of the generator of the diffusion. We investigate the system at criticality and show that the associated genealogical tree, when the process is conditioned to survive for a long time, converges to Aldous' Continuum Random Tree under appropriate rescaling. The result holds under only a mild assumption on the domain, and is valid for all branching mechanisms with finite variance, and a general class of diffusions.