A new approach to the single point catalytic super-Brownian motion |
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Authors: | Klaus Fleischmann Jean-François Le Gall |
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Institution: | (1) Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin, Germany;(2) Laboratoire de Probabilités, Université Pierre et Marie Curie, 4, Place Jussieu, Tour 56, F-75252 Paris Cedex 05, France |
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Abstract: | Summary A new approach is provided to the super-Brownian motionX with a single point-catalyst
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
c
ofX at the catalystc is distributed as the total occupation time measure ofU. Furthermore, we show thatX
t is determined from
c
by an explicit representation formula. Heuristically, a mass
c
(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
c
. In particular,x isC outside the catalyst. We also provide a new derivation of the singularity of the measure
c
. |
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Keywords: | 60J80 60J55 60G57 |
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