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.
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Fleischmann, K., Klenke, A. & Xiong, J. Pathwise Convergence of a Rescaled Super-Brownian Catalyst Reactant Process. J Theor Probab 19, 557–588 (2006). https://doi.org/10.1007/s10959-006-0025-2
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DOI: https://doi.org/10.1007/s10959-006-0025-2
Keywords
- Catalyst
- reactant
- superprocess
- martingale problem
- stochastic equation
- density field
- collision measure
- collision local time
- extinction
- critical scaling
- convergence in path space