Conditioning super-brownian motion on its boundary statistics; fragmentation and a class of weakly extreme x-harmonic functions

Abstract. LetX be a super-Brownian motion (SBM) defined on Rn and (XD) be its exit measures indexed by sub-domains of Rd. We pick a bounded sub-domainD, and condition the super-brownian motion inside this domain on its “boundary statistics”, random variables defined on an auxiliary probability space generated by sampling from the exit measure XD. Among these, two particular examples are: conditioning on a Poisson random measure with intensity βXD; and XD itself. We find the conditional laws as h-transforms of the original SBM law using Dynkin’s formulation of X-harmonic functions. For each conditioning we consider, we give an explicit expression for the corresponding X-harmonic function. We also obtain an explicit construction of some of these conditional laws in terms of branching particle systems. For example, we give a fragmentation system description of the law of super-Brownian motion conditioned on XD = ν, in terms of a particle system, called the backbone. In the backbone, each particle is labeled at its birth by a measure ν̃. The spatial motion of the particle is an h-transform of Brownian motion, where h is a potential that depends on ν̃. The label ν ̃ represents the particle and its descendants ’ total contribution to the exit measure. At the particle’s death two new particles are born and ν ̃ is passed to the newborns by fragmentation into two pieces. The X-harmonic function Hνx corresponding to conditioning on XD = ν is of special interest, as it can be thought as the analogue of the Poisson or Martin kernel. An open problem is to show that Hνx is extreme at least for some ν, when D is a smooth domain. An equivalent problem is to show the tail sigma field of SBM in D is trivial with respect to P νµ. We prove a weaker version of this result. We show that for any A in the tail sigma field of X, PXDµ (A) = 0 or 1. 1