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

The super-Brownian motion $X^\varrho$ in a super-Brownian medium $\varrho$ 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 $(d\leq3)$, 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 $(d\geq3)$, whereas in low dimensions the variances grow without bound, and the process clusters leading to local extinction