  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.

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