This thesis studies various problems related to the asymptotic behaviour and derivation of mean field models from systems of many particles.

Chapter 1 introduces mean field models and their derivation, and then summarises the following chapters of this thesis. Chapters 2, 3 and 4 directly study systems composed of many particles.

In Chapter 2 we prove quantitative propagation of chaos for systems of interacting SDEs with interaction kernels that are merely Hölder continuous (the usual assumption being Lipschitz). On the way we prove the existence of differentiable stochastic flows for a class of degenerate SDEs with rough coefficients and a uniform law of large numbers for SDEs.

Chapters 3 and 4 study the asymptotic behaviour of the Arrow-Hurwicz-Uzawa gradient method, which is a dynamical system for locating saddle points of concave-convex functions. This method is widely used in distributed optimisation over networks, for example in power systems and in rate control in communication networks. Chapter 3 gives an exact characterisation of the limiting solutions of the gradient method on the full space for arbitrary concave-convex functions. In Chapter 4 we extend this result to the subgradient method where the dynamics of the gradient method are restricted to an arbitrary convex set.

Chapters 5, 6 and 7 study the stability of mean field models. Chapters 5 and 6 prove an instability criterion for non-monotone equilibria of the Vlasov-Maxwell system. In Chapter 5 we study a related problem in approximation of the spectra of families of unbounded self adjoint operators. In Chapter 6 we show how the instability problem for Vlasov-Maxwell can be reduced to this spectral problem. In Chapter 7 we give a proof of well-posedness of a class of solutions to the Vlasov-Poisson system with unbounded spatial density. Chapters 8 and 9 change track and study the dynamics of a solute in a fluid background. In Chapter 8 we study a simple model for this phenomena, the kinetic Fokker-Planck equation, and show contraction of its semi-group in the Wasserstein distance when the spatial variable lies on the torus. Chapter 9 studies a more complex model of passive transport of a solute under a large and highly oscillatory fluid field. We prove a homogenisation result showing convergence to an effective diffusion equation for the transported solute profile.