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On the hot spots of a catalytic super-Brownian motion
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  • Published: November 2001

On the hot spots of a catalytic super-Brownian motion

  • Jean-François Delmas1 &
  • Klaus Fleischmann2 

Probability Theory and Related Fields volume 121, pages 389–421 (2001)Cite this article

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  • 3 Citations

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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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Authors and Affiliations

  1. MSRI, 1000 Centennial Drive, Berkeley, CA 94720, U.S.A. and ENPC-CERMICS, 6 av. Blaise Pascal, Champs-sur-Marne, 77455 Marne La Vallée, France. e-mail: delmas@cermics.enpc.fr, , , , , , FR

    Jean-François Delmas

  2. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr.39, D-10117 Berlin, Germany. e-mail: fleischmann@wias-berlin.de, , , , , , DE

    Klaus Fleischmann

Authors
  1. Jean-François Delmas
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  2. Klaus Fleischmann
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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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  • Issue Date: November 2001

  • DOI: https://doi.org/10.1007/s004400100156

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Keywords

  • Covariance
  • Local Time
  • Fixed Time
  • Hausdorff Dimension
  • Covariance Measure
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