WIAS Preprint No. 1052, (2005)

Hydrodynamic limit fluctuations of super-Brownian motion with a stable catalyst



Authors

  • Fleischmann, Klaus
  • Mörters, Peter
  • Wachtel, Vitali

2010 Mathematics Subject Classification

  • 60G57 60J80 60K35

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, Anderson model with stable random potential, infinite overall density

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 is an index associated to the medium.

Appeared in

  • Electron. J. Probab., 11 (2006) pp. 723-767.

Download Documents