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

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  • Ren, Yan-Xia

Abstract

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]

Suggested Citation

  • Ren, Yan-Xia, 2004. "Support properties of super-Brownian motions with spatially dependent branching rate," Stochastic Processes and their Applications, Elsevier, vol. 110(1), pages 19-44, March.
    • Handle: RePEc:eee:spapps:v:110:y:2004:i:1:p:19-44
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    References listed on IDEAS

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    1. Engländer, János & Fleischmann, Klaus, 2000. "Extinction properties of super-Brownian motions with additional spatially dependent mass production," Stochastic Processes and their Applications, Elsevier, vol. 88(1), pages 37-58, July.
    2. Sheu, Yuan-Chung, 1995. "On positive solutions of some nonlinear differential equations -- A probabilistic approach," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 43-53, September.
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