On the hot spots of a catalytic super-Brownian motion

Consider the catalytic super-Brownian motion Xϱ (reactant) in ℝd, d ≤ 3, which branching rates vary randomly in time and space and in fact are given by an ordinary super-Brownian motion ϱ (catalyst). Our main object of study is the collision local time L = L[ϱ,Xϱ](d[s,x]) of catalyst and reactant. It determines the covariance measure in the martingale problem for Xϱ and reflects the occurence of "hot spots" of reactant which can be seen in simulations of Xϱ. In dimension 2, spatial marginal collision densities exist and, via self-similarity, enter as factor in the long-term random ergodic limit of L (diffusiveness of the 2-dimensional model)