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

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.Comment: 40 page

Spread rate of catalytic branching symmetric stable processes

We study the growth order of the maximal displacement of branching symmetric $\alpha$-stable processes. We assume the branching rate measure $\mu$ is in the Kato class and $\mu$ has a compact support on ${\mathbb R}^d$. We show that the maximal displacement exponentially grows and its order is determined by the index $\alpha$ and the spectral bottom of the corresponding Schr\"odinger-type operator

Super-Brownian motion with extra birth at one point

A super-Brownian motion in two and three dimensions is constructed where "particles" give birth at a higher rate, if they approach the origin. Via a log-Laplace approach, the construction is based on Albeverio et al. (1995) who calculated the fundamental solutions of the heat equation with one-point potential in dimensions less than four