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Beschreibung:
Consider the continuous finite measure-valued super-Brownian motion X on ℝd corresponding to the evolution equation ut = ½Δ u + βu - u2, where β ∈ C γ (ℝd) with γ ∈ (0,1] is bounded from above. We prove criteria for (finite time) extinction and local extinction of X in terms of β. It turns out that for d ≤ 2, local extinction is equivalent with extinction. For general d, we show that if β has a suitable decay rate at infinity then it can be changed on a compact set in order to guarantee local extinction. On the other hand, if β is above this decay rate, the process does not exhibit local extinction. If d ≤ 6, then extinction has the same threshold rate as local extinction, while for d > 6 one observes a phase transition. Last, we show that in dimension 1, if β is unbounded from above and, in fact, degenerates to a single point source, then X does not exhibit local extinction, and the expectation of the rescaled process t ⟼ e-t/2Xt has a limit as t → ∞. In the proofs pde techniques and Laplace transforms are used together with h-transforms for measure-valued processes.