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

Abstract

We consider a critical finite measure-valued super-Brownian motion X=( X t , P μ ) in R d , whose log-Laplace equation is associated with the semilinear equation (∂/∂t)u= 1 2 Δu−ku 2 , 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 P μ ⋃ 0⩽s⩽t supp X s is bounded =1 for every t>0, and we say that the global support of X is compact if P μ ⋃ 0⩽s<∞ supp X 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)⩾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.