The First Hitting Manner of Super-Brownian Motions

Using the path construction of super-Brownian motions given by Le Gall, we investigate the first hitting manner of a domain by super-Brownian motions and show that the cardinal number of the intersection between closure of the domain and support of the super-Brownian motion at the first hitting time is less than 2. We also propose a conjecture that the cardinal number is at most one.