Journal of Theoretical Probability

, Volume 19, Issue 3, pp 557–588 | Cite as

Pathwise Convergence of a Rescaled Super-Brownian Catalyst Reactant Process

  • Klaus Fleischmann
  • Achim Klenke
  • Jie Xiong

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.


Catalyst reactant superprocess martingale problem stochastic equation density field collision measure collision local time extinction critical scaling convergence in path space 

Mathematics Subject classification 1991

Primary 60K35 Secondary 60G57 Secondary 60J80 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  2. 2.Institut für MathematikJohannes Gutenberg-Universität MainzMainzGermany
  3. 3.Department of MathematicsUniversity of TennesseeKnoxvilleUSA
  4. 4.Department of MathematicsHebei Normal UniversityShijiazhuangP.R. China

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