Absolute continuity of catalytic measure-valued branching processes

Abstract

Classical super-Brownian motion (SBM) is known to take values in the space of absolutely continuous measures only if d=1. For d⩾2 its values are almost surely singular with respect to Lebesgue measure. This result has been generalized to more general motion laws and branching laws (yielding different critical dimensions) and also to catalytic SBM. In this paper we study the case of a catalytic measure-valued branching process in R d with a Feller process ξ as motion process, where the branching rate is given by a continuous additive functional of ξ, and where also the (critical) branching law may vary in space and time. We provide a simple sufficient condition for absolute continuity of the values of this process. This criterion is sharp for the classical cases. As a partial converse we also give a sufficient condition for singularity of the states.