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Pathwise Convergence of a Rescaled Super-Brownian Catalyst Reactant Process

Author

Listed:
  • Klaus Fleischmann

    (Weierstrass Institute for Applied Analysis and Stochastics)

  • Achim Klenke

    (Johannes Gutenberg-Universität Mainz)

  • Jie Xiong

    (University of Tennessee
    Hebei Normal University)

Abstract

Consider the one-dimensional catalytic super-Brownian motion X (called the reactant) in the catalytic medium $$\varrho$$ which is an autonomous classical super-Brownian motion. We characterize $$(\varrho ,X)$$ both in terms of a martingale problem and (in dimension one) as solution of a certain stochastic partial differential equation. The focus of this paper is for dimension one the analysis of the longtime behavior via a mass-time-space rescaling. When scaling time by a factor of K, space is scaled by K η and mass by K −η. We show that for every parameter value η ≥ 0 the rescaled processes converge as K→ ∞ in path space. While the catalyst’s limiting process exhibits a phase transition at η = 1, the reactant’s limit is always the same degenerate process.

Suggested Citation

  • Klaus Fleischmann & Achim Klenke & Jie Xiong, 2006. "Pathwise Convergence of a Rescaled Super-Brownian Catalyst Reactant Process," Journal of Theoretical Probability, Springer, vol. 19(3), pages 557-588, December.
    • Handle: RePEc:spr:jotpro:v:19:y:2006:i:3:d:10.1007_s10959-006-0025-2
    DOI: 10.1007/s10959-006-0025-2
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    References listed on IDEAS

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    1. Dawson, Donald A. & Fleischmann, Klaus, 1988. "Strong clumping of critical space-time branching models in subcritical dimensions," Stochastic Processes and their Applications, Elsevier, vol. 30(2), pages 193-208, December.
    2. Donald A. Dawson & Klaus Fleischmann, 1997. "A Continuous Super-Brownian Motion in a Super-Brownian Medium," Journal of Theoretical Probability, Springer, vol. 10(1), pages 213-276, January.
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    Cited by:

    1. Hui He & Zenghu Li & Xiaowen Zhou, 2013. "An Integral Test on Time-Dependent Local Extinction for Super-coalescing Brownian Motion with Lebesgue Initial Measure," Journal of Theoretical Probability, Springer, vol. 26(1), pages 31-45, March.

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