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A new approach to the single point catalytic super-Brownian motion
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  • Published: 01 March 1995

A new approach to the single point catalytic super-Brownian motion

  • Klaus Fleischmann1 &
  • Jean-François Le Gall2 

Probability Theory and Related Fields volume 102, pages 63–82 (1995)Cite this article

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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Author information

Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117, Berlin, Germany

    Klaus Fleischmann

  2. Laboratoire de Probabilités, Université Pierre et Marie Curie, 4, Place Jussieu, Tour 56, F-75252, Paris Cedex 05, France

    Jean-François Le Gall

Authors
  1. Klaus Fleischmann
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  2. Jean-François Le Gall
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Cite this article

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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  • Received: 14 February 1994

  • Revised: 08 December 1994

  • Published: 01 March 1995

  • Issue Date: March 1995

  • DOI: https://doi.org/10.1007/BF01295222

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Mathematics Subject Classification

  • 60J80
  • 60J55
  • 60G57
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