Conditioning super-Brownian motion on its boundary statistics, and fragmentation

TL;DR

This paper studies how to condition super-Brownian motion on boundary-related statistics, providing explicit formulas and particle system constructions, including a fragmentation system description of conditioned laws.

Contribution

It introduces a novel framework for conditioning super-Brownian motion on boundary statistics using $h$-transforms and explicit constructions via branching particle systems.

Findings

Explicit $X$-harmonic functions for conditioned laws

Construction of conditioned super-Brownian motion via branching particle systems

Fragmentation system description of the conditioned process

Abstract

We condition super-Brownian motion on "boundary statistics" of the exit measure $X_D$ from a bounded domain $D$. These are random variables defined on an auxiliary probability space generated by sampling from the exit measure $X_D$. Two particular examples are: conditioning on a Poisson random measure with intensity $\beta X_D$ and conditioning on $X_D$ itself. We find the conditional laws as $h$-transforms of the original SBM law using Dynkin's formulation of $X$-harmonic functions. We give explicit expression for the (extended) $X$-harmonic functions considered. We also obtain explicit constructions of these conditional laws in terms of branching particle systems. For example, we give a fragmentation system description of the law of SBM conditioned on $X_D=\nu$, in terms of a particle system, called the backbone. Each particle in the backbone is labeled by a measure $\tilde{\nu}$, representing its descendants' total contribution to the exit measure. The particle's spatial motion is an $h$-transform of Brownian motion, where $h$ depends on $\tilde{\nu}$. At the particle's death two new particles are born, and $\tilde{\nu}$ is passed to the newborns by fragmentation.