Smooth density field of catalytic super-Brownian motion

Given an (ordinary) super-Brownian motion (SBM) . on Rd of dimension d=2,3, we consider a (catalytic) SBM X. on Rd with "local branching rates" .s(dx). We show that X.t is absolutely continuous with a density function ..t, say. Moreover, there exists a version of the map (t,z)...t(z) which is C. and solves the heat equation off the catalyst .; more precisely, off the (zero set of) closed support of the time-space measure ds.s(dx). Using self-similarity, we apply this result to give the following answer to an open problem on the long-term behavior of X. in dimension d=2: If . and X. start with a Lebesgue measure, then does X.T converge (persistently) as T.. toward a random multiple of Lebesgue measure?