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. We represent these laws in terms of “immortal particle” branching processes with immigration of mass, and relate them to the study of solutions to Lu=cu2 in D. The changes of measure include and generalize one arising by conditioning the support of the exit measure to hit a point z on the boundary of a 2-dimensional domain. In that case the branching process is the historical tree of the mass reaching z, and our results provide an explicit description of the law of this tree. In dimension 2 this conditioning is non-degenerate. The representations therefore differ from the related representations studied in an earlier paper, which treated the degenerate conditionings that arise in higher dimensions.