Absolute continuity of interacting measure-valued branching processes and its occupation-time processes

LetX _t be the interaction measured-valued branchingα-symmetric stable process overR _d (1< α ≤2) constructed by Meleard-Roelly^1]. Frist, it is shown thatX _t is absolutely continuous with respect to the Lebesgue measure (onR) with a continuous density function which satisfies some SPDE. Second, it is proved that if the underlying process is a Brownian motion onR ^d(d≤3), the corresponding occupation-time processY _t is also absolutely continuous with respect to the Lebesgue measure.