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 rates' #rho#_s(dx). We show that X_t"#rho# is absolutely continuous with a density function #xi#_t"#rho#, say. Moreover, there exists a version of the map (t, z)#->##xi#_t"#rho#(z) which is C"#infinity# and solves the heat equation of the catalyst #rho#, more precisely, of the (zero set of) closed support of the time-space measure ds #rho#_s(dx). Using self-similarity, we apply this result to answer the question of the long-term behavior of X"#rho# in dimension d=2: If #rho# and X"#rho# start with a Lebesgue measure, then X_T"#rho# converges (persistently) as T#->##infinity# towards a random multiple of Lebesgue measure. (orig.)Available from TIB Hannover: RR 5549(331)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman