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

We consider a critical finite measure-valued super-Brownian motion X=(Xt,P[mu]) in , whose log-Laplace equation is associated with the semilinear equation , where the coefficient k(x)>0 for the branching rate varies in space, and is continuous and bounded. Suppose that supp [mu] is compact. We say that X has the compact support property, if for every t>0, and we say that the global support of X is compact if . 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)[greater-or-equal, slanted]exp(-Mx2) as x-->[infinity] then X possesses the compact support property, whereas if there exist constant [beta]>2 such that k(x)[less-than-or-equals, slant]exp(-x[beta]) as x-->[infinity] then X does not have the compact support property. For the global support, we prove that if k(x)=x-[beta] (0[less-than-or-equals, slant][beta]<[infinity]) 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[greater-or-equal, slanted]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.