On States of Total Weighted Occupation Times for Superdiffusions

Abstract

Suppose X is a superdiffusion in Rd 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) = z1+β, 0 2 + 2/β, it is singular. As we know the absolute continuity and singularity of \( Y_{{\tau _{D} }} \) have not been discussed before.