We introduce Martingale changes of measure of the law of the exit measures of super Brownian motion from a sequence of domains. These changes of measure 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 three characterizations of the new process in terms of an "immortal particles" branching process with immigration of mass, and applications to the study of solutions to Lu = c u² in D. In an accompanying paper, we treat the degenerate case.

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 specified points on the boundary of a domain. The results are similar in flavor 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.

We construct a super iterated Brownian motion (SIBM) from a historical version of iterated Brownian motion (IBM) using an iterated Brownian snake (IBS). It is shown that the range of super iterated Brownian motion is qualitatively quite different from that of super Brownian motion in that there are points with explosions in the branching. However, at a fixed time the support of SIBM has an exact Hausdorff measure function that is the same (up to a constant) as that of super Brownian motion at a fixed time.

For the Brownian path-valued process of Le Gall (or Brownian snake) in \R², the times at which the process is a cone path are considered as a function of the size of the cone and the terminal position of the path. The results show that the paths for the path-valued process have local properties unlike those of a standard Brownian motion.