On States of Total Weighted Occupation Times for Superdiffusions

Suppose X is a superdiffusion in R~d with general branching mechanism ￠. and Ydenotesthe total weighted occupation time of X in a bounded smooth domain D. We discuss the conditions onψ to guarantee that Yhas absolutey continuous states. And for particular ψ(z) = z~(l+, 0<B ≤1.we prove that. in the case d<2 + 2/B. Ysup>D is absolutely continuous with respect to the Lebesguemeasure in D. whereas in the case d>2 + 2/B. it is singular. As we know the absolute continuity andsingularity of YD have not been discussed before.