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A Review of Recent Results on Hawkes Processes

Thursday, June 13, 2002 - 2:00pm - 2:50pm
Keller 3-180
Pierre Bremaud (École Polytechnique Fédérale de Lausanne (EPFL))
Such processes are also called branching point processes. They describe births times in a given colony as follows. There is a stationary point process of ancestors, born without parents, the events of which are the times of birth. The rest of the colony is is generated as follows. Call n a typical member born at time T(n). It has children according to a non-homogeneous Poisson process of intensity h(t-T(n),Z(n)), where Z(n) accounts for extra randomness in the model. Questions: Under what conditions is there a stationary point process with such dynamical description. If there is, is it unique, and how fast do we reach equilibrium, or extinction (in case the unique stationary solution is the empty process). Can we imagine such a process without ancestors (the answer is yes and this is of course elated to long-range dependence. What is the power spectrum (Bartlett spectrum) in the stationary case? All these questions have been to some extent also in the spatial case in a series of papers in collaboration with Laurent Massouli�, Gianluca Torrisi, and Gianna Nappo and I shall review these results, explaining them from the point of view of their potential interest in seismology.