Support properties of super-Brownian motions with spatially dependent branching rate

We consider a critical finite measure-valued super-Brownian motion X=(Xt,Pμ) in Rd, whose log-Laplace equation is associated with the semilinear equation (∂/∂t)u=12 Δu−ku2, where the coefficient k(x)>0 for the branching rate varies in space, and is continuous and bounded. Suppose that supp μ is compact. We say that X has the compact support property, if Pμ⋃0⩽s⩽t supp Xs is bounded=1 for every t>0, and we say that the global support of X is compact if Pμ⋃0⩽s<∞ supp Xs is bounded=1. We prove criteria for the compact support property and the compactness of the global support. If there exists a constant M>0 such that k(x)⩾exp(−M||x||2) as ||x||→∞ then X possesses the compact support property, whereas if there exist constant β>2 such that k(x)⩽exp(−||x||β) as ||x||→∞ then X does not have the compact support property. For the global support, we prove that if k(x)=||x||−β (0⩽β<∞) for sufficiently large ||x||, then the maximum decay order of k for the global support being compact is different for d=1, d=2 and d⩾3: it is O(||x||−3) in dimension one, O(||x||−2(log ||x||)−3) in dimension two, and O(||x||−2) in dimensions three or above.