Mean field for interacting particles subject to environmental noise

A system of interacting particles described by stochastic differential equations is considered. As opposed 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 (no interacting) 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 in viscid type, as opposed to the case when independent noises drive the different particles. Moreover, we use this result to derive a mean field approximation of the stochastic Euler equations for the vorticity of an incompressible fluid.