On the Conditioned Exit Measures of Super Brownian Motion

In this paper we present a martingale related to the exit measures of super Brow-nian 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 speciied 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. 1. Introduction One may think about super Brownian motion heuristically as a measure-valued process induced by a branching particle system in which each particle diiuses as a Brownian motion and critical branching is performed. With this image in mind, Dynkin's exit measure for super Brownian motion from a domain D is a measure on the boundary of D, supported on the set of points where the particles rst exit the domain. Le Gall proved in 19] that the exit measure will \hit" a xed point on the boundary of D, with positive probability, only in dimensions 1 and 2. In higher dimensions, a xed point on the boundary will be hit by an exiting particle with probability 0. One may ask, if we condition the process to hit that point, what do the paths of the particles that hit the point look like? A naive guess is to say they are Brownian particles which are conditioned to hit the point, and so should look like h-transforms of Brownian motion. In dimension 2, where no conditioning is necessary, this is essentially correct. Although the rst hit, as chronicled by the Brownian snake, is not an h-process, the typical path is. This has been exploited by Abraham 1].-page 1