Electronic Journal of Probability

Hydrodynamic Limit Fluctuations of Super-Brownian Motion with a Stable Catalyst

Klaus Fleischmann, Peter Mörters, and Vitali Wachtel

Full-text: Open access

Abstract

We consider the behaviour of a continuous super-Brownian motion catalysed by a random medium with infinite overall density under the hydrodynamic scaling of mass, time, and space. We show that, in supercritical dimensions, the scaled process converges to a macroscopic heat flow, and the appropriately rescaled random fluctuations around this macroscopic flow are asymptotically bounded, in the sense of log-Laplace transforms, by generalised stable Ornstein-Uhlenbeck processes. The most interesting new effect we observe is the occurrence of an index-jump from a Gaussian situation to stable fluctuations of index $1+\gamma$, where $\gamma \in (0,1)$ is an index associated to the medium.

Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 29, 723-767.

Dates
Accepted: 27 August 2006
First available in Project Euclid: 31 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464730564

Digital Object Identifier
doi:10.1214/EJP.v11-348

Mathematical Reviews number (MathSciNet)
MR2242662

Zentralblatt MATH identifier
1110.60082

Subjects
Primary: 60G57: Random measures
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Catalyst reactant superprocess critical scaling refined law of large numbers catalytic branching stable medium random environment supercritical dimension generalised stable Ornstein-Uhlenbeck process index jump parabolic Anderson model with sta

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Fleischmann, Klaus; Mörters, Peter; Wachtel, Vitali. Hydrodynamic Limit Fluctuations of Super-Brownian Motion with a Stable Catalyst. Electron. J. Probab. 11 (2006), paper no. 29, 723--767. doi:10.1214/EJP.v11-348. https://projecteuclid.org/euclid.ejp/1464730564


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