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.
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Dawson, D.A., Fleischmann, K. A Continuous Super-Brownian Motion in a Super-Brownian Medium. Journal of Theoretical Probability 10, 213–276 (1997). https://doi.org/10.1023/A:1022606801625
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DOI: https://doi.org/10.1023/A:1022606801625
- 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