Hydrodynamic limit fluctuations of super-Brownian motion with a stable catalyst
Klaus Fleischmann, Weierstrass Institute for Applied Analysis and Stochastics, Berlin
Peter Mörters, University of Bath
Vitali Wachtel, Weierstrass Institute for Applied Analysis and Stochastics, Berlin
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+γ, where γ ∈ (0,1)
is an index associated to the medium.
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