  We propose a framework for studying predictability of extreme events in
complex systems. Major conceptual elements -- hierarchical structure, spatial
dynamics, and external driving -- are combined in a classical branching
diffusion with immigration. New elements -- observation space and observed
events -- are introduced in order to formulate a prediction problem patterned
after the geophysical and environmental applications. The problem consists of
estimating the likelihood of occurrence of an extreme event given the
observations of smaller events while the complete internal dynamics of the
system is unknown. We look for premonitory patterns that emerge as an extreme
event approaches; those patterns are deviations from the long-term system's
averages. We have found a single control parameter that governs multiple
spatio-temporal premonitory patterns. For that purpose, we derive i) complete
analytic description of time- and space-dependent size distribution of
particles generated by a single immigrant; ii) the steady-state moments that
correspond to multiple immigrants; and iii) size- and space-based asymptotic
for the particle size distribution. Our results suggest a mechanism for
universal premonitory patterns and provide a natural framework for their
theoretical and empirical study.
