A continuous super-Brownian motion in a super-Brownian medium
- Dawson, Donald A.
- Fleischmann, Klaus
2010 Mathematics Subject Classification
- 60J80 60J55 60G57
- catalytic reaction diffusion equation, catalyst process, random medium, catalytic medium, super-Brownian motion, superprocess, branching rate functional, measure-valued branching, critical branching, occupation time, jointly continuous occupation density, Hölder continuities, collision local time, persistence, super-Brownian medium
A continuous super-Brownian motion Xᵨ is constructed in which branching occurs only in the presence of catalysts which evolve themselves as a continuous super-Brownian motion ᵨ. More precisely, the collision local time L[W,ᵨ] (in the sense of Barlow et al. [BEP91]) of an underlying Brownian motion path W with the catalytic mass process ᵨ governs 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 positive (finite) limiting variance, whereas starting with a Lebesgue measure ℓ, stochastic convergence to ℓ occurs.
- J. Theoret. Probab., 10 (1997), pp. 213-276