Smooth density field of catalytic super-Brownian motion

Given an (ordinary) super-Brownian motion (SBM) rho on R-d of dimension d = 2, 3, we consider a (catalytic) SBM X-rho on R-d with "local branching rat es" rho(s)(dx). We show that X-t(rho) is absolutely continuous with a densi ty function xi(t)(rho), say. Moreover, there exists a version of the map (t , z) --> xi(t)(rho)(z) which is l(infinity) and solves the heat equation of f the catalyst rho; more precisely, off the (zero set of) closed support of the time-space measure ds rho(s)(dx). Using self-similarity we apply this result to give the following answer to an open problem on the long-term beh avior of X-rho in dimension d = 2: if rho and X-rho Start with a Lebesgue m easure, then does X-T(rho) converge (persistently) as T --> infinity toward a random multiple of Lebesgue measure?