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

The super-Brownian motion Xϱ in a catalytic medium ϱ constructed in [DF96a] is known to be persistent (no loss of expected mass in the longtime behaviour) in dimensions one ([DF96a]) and three ([DF96b]). 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