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

Abstract

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 βXD and conditioning on XD itself. We find the conditional laws as htransforms 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 = ν, in terms of a particle system, called the backbone. Each particle in the backbone is labeled by a measureν, representing its descendants' total contribution to the exit measure. The particle's spatial motion is an h-transform of Brownian motion, where h depends onν. At the particle's death two new particles are born, andν is passed to the newborns by fragmentation.

Moment Densities of Super Brownian Motion, and a Harnack Estimate for a Class of X-harmonic Functions

“…ν, so to proceed further we will need to specify more regular versions. The following is contained in Theorem 2 of Salisbury and Sezer (2012) (whose proof is a modification of that of Theorem 1.1 of Dynkin (2006b)), and establishes the existence of a jointly measurable version of H ν (µ) that is extended X-harmonic for R-almost all ν. We assume that D is a bounded domain all of whose boundary points are regular.…”

“…Because H is X-harmonic, the Markov property implies that these probability laws will be consistent as we vary D ′ , so can be uniquely extended to F D− = σ(X D ′ , D ′ ⋐ D), as in Section 1.3 of Dynkin (2006b). See also Theorem 2(d) of Salisbury and Sezer (2012). Note that the mutual absolute continuity of P µ1,XD and P µ2,XD , for µ 1 and µ 2 compactly supported in D, implies that if H(µ) is non-zero from one µ then it is non-zero for all µ.…”

“…The densities H ν D played a key role in Salisbury and Sezer (2012) as well, in particular, in the analysis of super-Brownian motion (X D ′ ) D ′ ⋐D conditioned on its exit measure X D . In addition to Theorem 2, that paper showed that P is the conditional law of SBM given X D = ν, for R-a.e.…”

“…In the current paper we will use the representation of Salisbury and Sezer (2012) to tie the densities H ν D to the functions ρ(x, z , . .…”

Change of measure in the lookdown particle system

“…For fixed s ≥ 0, the additive h-transform of the GFV process on [0, s] may be obtained by conditioning a random particle chosen at time t, t large, to move as an h-transform. For other conditionings on boundary statistics in the context of measure valued branching processes, we refer to Salisbury and Sezer [18]. (e −λu −1 + λu1 u≤1 )ν Y (du), (21) for ν Y a Lévy measure such that (0,∞) (1 ∧ u 2 )ν Y (du) < ∞, β ∈ R, and σ 2 ∈ R + .…”