A Continuous Super-Brownian Motion in a Super-Brownian Medium

A continuous super-Brownian motion XQ is constructed in which branching occurs only in the presence of catalysts which evolve themselves as a continuous super-Brownian motion Q . More precisely, the collision local time L[W,Q] (in the sense of Barlow et al.) of an underlying Brownian motion path W with the catalytic mass process Q goerns the branching (in the sense of Dynkin's additive functional approach). In the one-dimensional case, a new type of limit behavior is encountered: The total mass process converges to a limit without loss of expectation mass (persistence) and with a nonzero limiting variance, whereas starting with a Lebesgue measure ℓ , stochastic convergence to ℓ occurs