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Extinction properties of super-Brownian motions with additional spatially dependent mass production

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  • Engländer, János
  • Fleischmann, Klaus

Abstract

Consider the finite measure-valued continuous super-Brownian motion X on corresponding to the log-Laplace equation where the coefficient [beta](x) for the additional mass production varies in space, is Hölder continuous, and bounded from above. We prove criteria for (finite time) extinction and local extinction of X in terms of [beta]. There exists a threshold decay rate kdx-2 as x-->[infinity] such that X does not become extinct if [beta] is above this threshold, whereas it does below the threshold (where for this case [beta] might have to be modified on a compact set). For local extinction one has the same criterion, but in dimensions d>6 with the constant kd replaced by Kd>kd (phase transition). h-transforms for measure-valued processes play an important role in the proofs. We also show that X does not exhibit local extinction in dimension 1 if [beta] is no longer bounded from above and, in fact, degenerates to a single point source [delta]0. In this case, its expectation grows exponentially as t-->[infinity].

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:88:y:2000:i:1:p:37-58
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    References listed on IDEAS

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    1. Sheu, Yuan-Chung, 1997. "Lifetime and compactness of range for super-Brownian motion with a general branching mechanism," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 129-141, October.
    2. Dawson, Donald A. & Fleischmann, Klaus, 1994. "A super-Brownian motion with a single point catalyst," Stochastic Processes and their Applications, Elsevier, vol. 49(1), pages 3-40, January.
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    Cited by:

    1. Grummt, Robert & Kolb, Martin, 2013. "Law of large numbers for super-Brownian motions with a single point source," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1183-1212.
    2. 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.

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