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

We consider a critical finite measure-valued super-Brownian motion X = (X-t,P-mu) in R-d, log-Laplace equation is associated with the semilinear equation (partial derivative/partial derivativet)u = 1/2 Deltau - ku(2), 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 P-mu (U-0less than or equal to8less than or equal to1 supp X-s is bounded) = 1 for every t > 0, and we say that the global support of X is compact if P-mu (U-0less than or equal tosless than or equal toinfinity suppX(s) 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) greater than or equal to exp(-M\\x\\(2)) as \\x\\ --> infinity then X possesses the compact support property, whereas if there exist constant beta > 2 such that k(x) less than or equal to 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)=parallel toxparallel to(-beta) (0 less than or equal to beta < infinity) for sufficiently large parallel toxparallel to, then the maximum decay order of k for the global support being compact is different for d = 1, d = 2 and d greater than or equal to 3: it is O(parallel toxparallel to(-3)) in dimension one, O(parallel toxparallel to(-2)(log parallel toxparallel to)(-3)) in dimension two, and O(parallel toxparallel to(-2)) in dimensions three or above. (C) 2003 Elsevier B.V. All rights reserved.Statistics & ProbabilitySCI(E)3ARTICLE119-4411