Abstract.
Consider the catalytic super-Brownian motion X ϱ (reactant) in ℝd, 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 themartingale 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 s ϱ have carrying Hausdorff dimension 2. Spatial marginal densities of L exist, and, via self-similarity, enter in the long-term randomergodic limit of L (diffusiveness of the 2-dimensional model). We alsocompare some of our results with the case of super-Brownian motions withdeterministic time-independent catalysts.
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Received: 2 December 1998 / Revised version: 2 February 2001 / Published online: 9 October 2001
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Delmas, JF., Fleischmann, K. On the hot spots of a catalytic super-Brownian motion. Probab Theory Relat Fields 121, 389–421 (2001). https://doi.org/10.1007/s004400100156
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DOI: https://doi.org/10.1007/s004400100156
Keywords
- Covariance
- Local Time
- Fixed Time
- Hausdorff Dimension
- Covariance Measure