Conditioning super-Brownian motion on its boundary statistics and fragmentation

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 1 (nu) over tilde, 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 (nu) over tilde. At the particle's death two new particles are born, and ($) over tilde is passed to the newborns by fragmentation.