Space-time regularity of catalytic super-Brownian motion

We study the question for which catalysts does the catalytic super-Brownian motion in R have a jointly continuous space-time Lebesgue density. As it turns out, there is a large class of non-atomic catalysts inducing a regular density. The latter can be characterized as the unique solution to a certain stochastic partial differential equation driven by an inhomogeneous space-time white noise