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Branching processes with competition by pruning of Levy trees

Wednesday 18th March 2015 - 15:30 to 16:30
INI Seminar Room 1
Co-authors: Joaquin Fontbona (U. Chile), Maria Clara Fittipaldi (U. Chile), L. Doering (U. Zurich), L. Mytnik (Technion), L. Zambotti (UPMC)

There are several ways to describe the evolution of a population with no interactions between individuals. One approach is to use the local time process of a forrest of Lévy trees, or, following the work of Dawson and Li, one can construct the whole population flow as the solution to a certain system of Lévy driven stochastic differential equation. The equivalence between these two constructions is a generalization of the well-known Ray-Knight Theorem.

When one wants to introduce a form of competition in the population, the situation becomes more involved. The stochastic differential approach still works (with an added negative drift term) and the purpose of this talk is to present a novel construction based on the interactive pruning of the Lévy forrest.

The case of a positive drift, which corresponds to an interactive immigration, is also of interest as it is related to the question of existence of exceptional times for Generalized Fleming-Viot processes with mutations at which the number of genetic types in the population is finite.

Based on joint works with : a) L. Doering, L. Mytnik and L. Zambotti and b) J. Fontbona and M.C. Fittipaldi

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons