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

Abstract

A continuous super-Brownian motion $$X^Q $$ 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. (1)) 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 $$\ell$$ , stochastic convergence to $$\ell$$ occurs.