Non-degenerate Conditionings of the Exit Measures of Super Brownian Motion

We introduce several martingale changes of measure of the law of the exit measure of super Brownian motion. These changes of measure include and generalize one arising by conditioning the exit measures to charge a point on the boundary of a 2-dimensional domain. In the case we discuss this is a non-degenerate conditioning. We give characterizations of the new processes in terms of \immortal particle" branching processes with immigration of mass, and give applications to the study of solutions to Lu = cu 2 in D. The representations are related to those in an earlier paper, which treated the case of degenerate conditionings. 1. Introduction We investigate conditionings of the exit measures of super Brownian motionin R d. We can think of super Brownian motion as the limit of a particle system, which can heuristically be described as follows. It consists of a cloud of particles, each diiusing as a Brownian motion and undergoing critical branching. A measure valued process is formed by assigning a small point mass to each particle's position at a given time. The exit measure X D from a domain D is then formed by freezing this mass at the point the particle rst exits from D. For a sequence of subdomains, these measures can be deened on the same probability space, giving rise to a process indexed by the subdomains. In dimension 2, with positive probability, points on the boundary of a smooth enough domain will be hit by the support of the exit measure. In this paper, we study conditionings of the sequence of exit measures, analogous to the conditioning by this event. Unlike the case d = 2, in higher dimensions the corresponding event has probability 0, and the analogous conditioning is a degenerate one. Such degenerate conditionings were treated in the paper 12] (SV1). To be more speciic, let D be a bounded domain in dimension d = 2, and let D k be an increasing sequence of subdomains. The domains D k give rise to a process of exit measures X k , each deened on the boundary of D k. We work under N x , the excursion measure under-page 1