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Persistence of a two-dimensional super-Brownian motion in a catalytic medium
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  • Published: January 1998

Persistence of a two-dimensional super-Brownian motion in a catalytic medium

  • Alison M. Etheridge1 &
  • Klaus Fleischmann2 

Probability Theory and Related Fields volume 110, pages 1–12 (1998)Cite this article

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  • 9 Citations

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Summary.

The super-Brownian motion X ϱ in a super-Brownian medium ϱ constructed in [DF97a] is known to be persistent (no loss of expected mass in the longtime behaviour) in dimensions one ([DF97a]) and three ([DF97b]). Here we fill the gap in showing that persistence holds also in the critical dimension two. The key to this result is that in any dimension (d≤3), given the catalyst, the variance of the process is finite `uniformly in time'. This is in contrast to the `classical' super-Brownian motion where this holds only in high dimensions (d≥3), whereas in low dimensions the variances grow without bound, and the process clusters leading to local extinction.

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Authors and Affiliations

  1. School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, UK. e-mail: A.M.Etheridge@qmw.ac.uk, , , , , , GB

    Alison M. Etheridge

  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. Alison M. Etheridge
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  2. Klaus Fleischmann
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Received: 21 November 1996 / In revised form: 31 March 1997

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Etheridge, A., Fleischmann, K. Persistence of a two-dimensional super-Brownian motion in a catalytic medium. Probab Theory Relat Fields 110, 1–12 (1998). https://doi.org/10.1007/s004400050142

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  • Issue Date: January 1998

  • DOI: https://doi.org/10.1007/s004400050142

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  • AMS Subject Classification (1991): Primary 60 J80; Secondary 60 G57
  • 60 K35
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