On the hot spots of a catalytic super-Brownian motion

Consider the catalytic super-Brownian motion X^ρ (reactant) in ℝ^d, 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 X^ρ and reflects the occurrence of "hot spots" of reactant which can be seen in simulations of X^ρ. In dimension 2, the collision local time is absolutely continuous in time, L(d(s,x)) = ds K_s(dx). At fixed time s, the collision measures K_s(dx) of ρ_s and X^ρ_s have carrying Hausdorff dimension 2. Spatial marginal densities of L exist, and, via self-similarity, enter in the long-term random ergodic limit of L (diffusiveness of the 2-dimensional model). We also compare some of our results with the case of super-Brownian motions with deterministic time-independent catalysts.