Interacting particle systems, kinetic equations and mean field games

Research Project, 2013 – 2016

The project deals with several aspects of interacting many particle systems, and in particular the limits when the number of particles approach infinity. A key concept is propagation of chaos. It means essentially that two particles entering a collision do not have a common history, and thus can be considered as being independent. This was assumed to hold in Boltzmann´s derivation of the equation that bears his name, but it was given a precise mathematical formulation much later. One of the main objectives with the project is to investigate condistions under which propagation of chaos holds for several stochastic interacting particle systems. The second main theme is that of mean field game theory. This is a technique for deriving and analysing efficient equations to describe mathematical games with a very large number of players. Also here we are mostly intersted in the limit of infinitely many players, and although there are significant differences with the stochastic particle systems considered in the first part of the project, the mathematical techniques used are similar, and the two directions of research will be of use for each other. Many of the interacting particle systems that we will consider in the project are taken from biological models: swarming animals and certain games used to describe the social behaviour of animals are some examples of this.

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