For critical spatially homogeneous branching processes of finite intensity the following dichotomy is well-known: convergence to non-trivial steady states, or local extinction. In the latter case the underlying phenomenon is the growth of large clumps at spatially rare sites. For this situation a precise description is given in terms of a scaling limit theorem provided that the dimension of the ambient space is small enough. In fact, a space-time-mass scaling limit exists and is a critical measure-valued branching process without a motion component. The clumps are located at Poissonian points and their sizes evolve according to critical continuous-state Galton-Watson processes. The spatial irregularities (intermittency) will grow in the sense that clumps will disappear as time increases in spite of the fact that the overall density remains constant in time.

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Keywords branching with infinite variance, clumping, intermittency, measureivalued branching, non-linear diffusion equation, scaling limit theorem, self-similarity, stable flow
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Journal Stochastic Processes and their Applications
Dawson, D.A, & Fleischmann, K. (Klaus). (1988). Strong clumping of critical space-time branching models in subcritical dimensions. Stochastic Processes and their Applications, 30(2), 193–208. doi:10.1016/0304-4149(88)90084-1