Local extinction for superprocesses in random environments

We consider a superprocess in a random environment represented by a random measure which is white in time and colored in space with correlation kernel g(x, y). Suppose that g(x, y) decays at a rate of |x − y|^−α 0 ≤ α ≤ 2, as |x − y| → ∞. We show that the process, starting from Lebesgue measure, suffers longterm local extinction. If 0 ≤ α ≤ 2, then it even suffers finite time local extinction. This property is in contrast with the classical super-Brownian motion which has a non-trivial limit when the spatial dimension is higher than 2. We also show in this paper that in dimensions d = 1, 2 superprocess in random environment suffers local extinction for any bounded function g.