On States of Total Weighted Occupation Times for Superdiffusions

Abstract

Suppose X is a superdiffusion in R^d with general branching mechanism ψ, and $Y_{{\tau _{D} }}$ denotes the total weighted occupation time of X in a bounded smooth domain D. We discuss the conditions on ψ to guarantee that $Y_{{\tau _{D} }}$ has absolutely continuous states. And for particular ψ ₍ z) = z ^1+β, 0 < β≤ 1, we prove that, in the case d < 2 + 2/β, $Y_{{\tau _{D} }}$ is absolutely continuous with respect to the Lebesgue measure in ̄D̄, whereas in the case d > 2 + 2/β, it is singular. As we know the absolute continuity and singularity of $Y_{{\tau _{D} }}$ have not been discussed before.