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

We condition super-Brownian motion on .boundary statistics. of the exit measure XD from a bounded domain D. These are random variables defined on an auxiliary probability space generated by sampling from the exit measure XD. Two particular examples are: conditioning on a Poisson random measure with intensity .XD and conditioning on XD 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 XD=., in terms of a particle system, called the backbone. Each particle in the backbone is labeled by a measure ~., 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 ~.. At the particle.s death two new particles are born, and ~. is passed to the newborns by fragmentation.