Random many-particle systems: applications from biology, and propagation of chaos in abstract models

Abstract

The paper discusses a family of Markov processes that represent many particle systems, and their limiting behaviour when the number of particles go to infinity. The first part concerns model of biological systems: a model for sympatric speciation, i.e. the process in which a genetically homogeneous population is split in two or more different species sharing the same habitat, and models for swarming animals. The second part of the paper deals with abstract many particle systems, and methods for rigorously deriving mean field models.Comment: These are notes from a series of lectures given at the 5$^{th}$ Summer School on Methods and Models of Kinetic Theory, Porto Ercole, 2010. They are submitted for publication in "Rivista di Matematica della Universit\`a di Parma