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September 2013 Conditioning super-Brownian motion on its boundary statistics, and fragmentation
Thomas S. Salisbury, A. Deniz Sezer
Ann. Probab. 41(5): 3617-3657 (September 2013). DOI: 10.1214/12-AOP778


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 $\tilde{\nu}$, 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.


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Thomas S. Salisbury. A. Deniz Sezer. "Conditioning super-Brownian motion on its boundary statistics, and fragmentation." Ann. Probab. 41 (5) 3617 - 3657, September 2013.


Published: September 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1296.60224
MathSciNet: MR3127894
Digital Object Identifier: 10.1214/12-AOP778

Primary: 60J25 , 60J60
Secondary: 60J80

Keywords: $X$-harmonic functions , branching backbone system , conditioning super-Brownian motion , diffusion , extreme $X$-harmonic functions , fragmentation , Martin boundary , measure valued processes , Poisson random measure

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 5 • September 2013
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