Mean field limit for interacting particles and filaments

In this thesis several systems of interacting particles are considered. The manuscript is divided in three parts. The first two parts are dedicated to the study of stochastic point particles, in the last part a system of interacting filaments is considered. As opposed to the usual model, where the noise perturbations acting on different particles are independent, the particles studied in the first part are subject to the same space-dependent noise. A result of propagation of chaos is proved and it is shown that the limit PDE is stochastic and of inviscid type, as opposed to the case when independent noises drive the different particles. In the second part a system similar as before is considered. Here the limit equation is not regular anymore but it is the stochastic Euler equation for the vorticity of a fluid with multiplicative noise. A mean field approximation of these equation is proved, where the particles are subject to a regularized interaction. In the last part, families of interacting curves are considered, with long range, mean field type interaction. A family of curves defines a 1-current analog to the empirical measure of interacting point particles. This current is proved to converge to a mean field current, solution of a nonlinear, vector valued, partial differential equation. This set-up is inspired from vortex filaments in turbulent fluids, but here a smooth interaction is considered, instead of the singular Biot-Savart kernel.