Finite time extinction of super-Brownian motions with deterministic catalyst

Abstract

In this paper we consider a super-Brownian motion X with branching mechanism k(x)z α, where k(x) > 0 is a bounded Holder continuous function on ℝd and\({}_{{}_x \in \mathbb{R}^d }k(x) = 0\). We prove that ifk(x)⩾‖x‖−l(0⩽l<∞) for sufficiently large x, then X has compact support property, and for dimension d = 1, ifk(x)⩾exp(=−l‖x‖)(0⩽l<∞) for sufficiently large x, then X also has compact support property. The maximal order of k(x) for finite time extinction is different between d = 1, d = 2 and d ≥ 3: it isO(‖x‖−(α+1)) in one dimension,O(‖x‖−2(log ‖x‖))−(α+1)) in two dimensions, andO(‖x‖2) in higher dimensions. These growth orders also turn out to be the maximum order for the nonexistence of a positive solution for 1/2Δu=k(x)uα.