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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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