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