## Conditioning super-Brownian motion on its boundary statistics, and fragmentation.(English)Zbl 1296.60224

Authors’ 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 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.

### MSC:

 60J68 Superprocesses 60J25 Continuous-time Markov processes on general state spaces 60J60 Diffusion processes 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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### References:

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