A system of interacting particles described by stochastic differential equations is considered. As oppopsed to the usual model, where the noise perturbations acting on different particles are independent, here the particles are subject to the same space-dependent noise, similar to the (noninteracting) particles of the theory of diffusion of passive scalars. We prove a result of propagation of chaos and show that the limit PDE is stochastic and of inviscid type, as opposed to the case when independent noises drive the different particles.

We take a pathwise approach to classical McKean-Vlasov stochastic differential equations with additive noise, as e.g. exposed in Sznitmann [38]. Our study was prompted by some concrete problems in battery modelling [23], and also by recent progrss on rough-pathwise McKean-Vlasov theory, notably Cass-Lyons [10], and then Bailleul, Catellier and Delarue [4]. Such a "pathwise McKean-Vlasov theory" can be traced back to Tanaka [40]. This paper can be seen as an attempt to advertize the ideas, power and simplicity of the pathwise appproach, not so easily extracted from [4,10,40], together with a number of novel applications. These include mean field convergence without a priori independence and exchangeability assumption; common noise, càdlàg noise, and reflecting boundaries. Last not least, we generalize Dawson-Gärtner large deviations and the central limit theorem to a non-Brownian noise setting.

Motivated by the challenge of incorporating data into misspecified and multiscale dynamical models, we study a McKean-Vlasov equation that contains the data stream as a common driving rough path. This setting allows us to prove well-posedness as well as continuity with respect to the driver in an appropriate rough-path topology. The latter property is key in our subsequent development of a robust data assimilation methodology: We establish propagation of chaos for the associated interacting particle system, which in turn is suggestive of a numerical scheme that can be viewed as an extension of the ensemble Kalman filter to a rough-path framework. Finally, we discuss a data-driven method based on subsampling to construct suitable rough path lifts and demonstrate the robustness of our scheme in a number of numerical experiments related to parameter estimation problems in multiscale contexts.

Families of N interacting curves are considered, with long range, mean field type, interaction. A family of curves defines a 1-current, concentrated on the curves, analog of the empirical measure of interacting point particles. This current is proved to converge, as N goes to infinity, to a mean field current, solution of a nonlinear, vector valued, partial differential equation. In the limit, each curve interacts with the mean field current and two different curves have an independence property if they are independent at time zero. This set-up is inspired from vortex filaments in turbulent fluids, although for technical reasons we have to restrict to smooth interaction, instead of the singular Biot-Savart kernel. All these results are based on a careful analysis of a nonlinear flow equation for 1-currents, its relation with the vector valued PDE and the continuous dependence on the initial conditions.

The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker-Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting diffusions with common noise. The uniqueness of solutions is obtained without any higher moment assumption on the solution by means of a duality argument to a backward stochastic PDE.