Pathwise Convergence of a Rescaled Super-Brownian Catalyst Reactant Process

Klaus Fleischmann (), Achim Klenke () and Jie Xiong ()
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Klaus Fleischmann: Weierstrass Institute for Applied Analysis and Stochastics
Achim Klenke: Johannes Gutenberg-Universität Mainz
Jie Xiong: University of Tennessee

Journal of Theoretical Probability, 2006, vol. 19, issue 3, 557-588

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.

Keywords: Catalyst; reactant; superprocess; martingale problem; stochastic equation; density field; collision measure; collision local time; extinction; critical scaling; convergence in path space; Primary 60K35; Secondary 60G57; Secondary 60J80 (search for similar items in EconPapers)
Date: 2006
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DOI: 10.1007/s10959-006-0025-2

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