Summary
A new approach is provided to the super-Brownian motionX with a single point-catalyst δ c as branching rate. We start from a superprocessU with constant branching rate and spatial motion given by the 1/2-stable subordinator. We prove that the occupation density measure λc ofX at the catalystc is distributed as the total occupation time measure ofU. Furthermore, we show thatX t is determined from λc by an explicit representation formula. Heuristically, a mass λc(ds) of “particles” leaves the catalyst at times and then evolves according to Itô's Brownian excursion measure. As a consequence of our representation formula, the density fieldx ofX satisfies the heat equation outside ofc, with a noisy boundary condition atc given by the singularly continuous random measure λc. In particular,x isC outside the catalyst. We also provide a new derivation of the singularity of the measure λc.
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Fleischmann, K., Le Gall, JF. A new approach to the single point catalytic super-Brownian motion. Probab. Theory Relat. Fields 102, 63–82 (1995). https://doi.org/10.1007/BF01295222
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DOI: https://doi.org/10.1007/BF01295222