On the conditioned exit measures of super Brownian motion

Abstract

Summary:In this paper we present a martingale related to the exit measures of super Brownian motion. By changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application is given to the problem of conditioning the exit measure to hit a number of speci ed points on the boundary of a domain. The results are similar in avor to the \immortal particle" picture of conditioned super Brownian motion but more general, as the change of measure is given by a martingale which need not arise from a single harmonic function.running head: Exit measures

The Backbone Decomposition for Superprocesses with Non-local Branching

“…The so-called backbone decompositions have been known in the earlier and more analytical setting of semigroup decompositions through the work of [8] and [7] as well as in the pathwise setting in the work of [14,15]. The purpose of this section is to introduce the pathwise backbone decomposition for a supercritical superprocess without non-local branching given in [1], we hope this will make the rest of the paper easier to follow.…”

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

“…and for general C we have the following recursive formulas; see, for example, Theorem 5.1.1 of Dynkin (2004), or Lemma 2.6 of Salisbury and Verzani (1999):…”

“…At the particle's death two new particles are born, and ν is passed to the newborns by fragmentation into two bits. Here, we use the techniques of Salisbury and Verzani (1999) applied to a more general setting. This description connects the theory of conditioned super-processes to the growing literature on infinite fragmentation and coalescent processes; see, for example, Bertoin (2006) for a comprehensive exposition.…”

A class of extreme X-harmonic functions

“…The X-harmonic functions H g,h 1 ,...,h N . In the papers [12] and [11] a class of X-harmonic functions was introduced that arose in the study of conditioning the superprocess to behave in a certain way as it exited E. To describe the process we need to review killing and transformations of processes.…”