Blowup and Conditionings of $\psi$-super Brownian Exit Measures

TL;DR

This paper extends conditioning results for super-Brownian motion to general branching rules, revealing new structures in the conditioned process and analyzing singularities in stable branching cases.

Contribution

It introduces representations of conditioned super-Brownian motion as an $h$-transform and as a superprocess with immigration, highlighting differences from finite-variance branching models.

Findings

Conditioned processes can be represented as $h$-transforms and superprocesses with immigration.

Stable branching leads to singularities and non-binary branching trees.

Mass creation at branch points can be strictly positive, unlike finite-variance cases.

Abstract

We extend earlier results on conditioning of super-Brownian motion to general branching rules. We obtain representations of the conditioned process, both as an $h$-transform, and as an unconditioned superprocess with immigration along a branching tree. Unlike the finite-variance branching setting, these trees are no longer binary, and strictly positive mass can be created at branch points. This construction is singular in the case of stable branching. We analyze this singularity first by approaching the stable branching function via analytic approximations. In this context the singularity of the stable case can be attributed to blow up of the mass created at the first branch of the tree. Other ways of approaching the stable case yield a branching tree that is different in law. To explain this anomaly we construct a family of martingales whose backbones have multiple limit laws.