The Annals of Probability

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

Thomas S. Salisbury and A. Deniz Sezer

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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.

Article information

Ann. Probab., Volume 41, Number 5 (2013), 3617-3657.

First available in Project Euclid: 12 September 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 60J60: Diffusion processes [See also 58J65]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

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


Salisbury, Thomas S.; Sezer, A. Deniz. Conditioning super-Brownian motion on its boundary statistics, and fragmentation. Ann. Probab. 41 (2013), no. 5, 3617--3657. doi:10.1214/12-AOP778.

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