Asymptotic behaviour for interacting diffusion processes with space-time random birth

We study the asymptotic behaviour of a system of interacting particles with space-time random birth. We have propagation of chaos and obtain the convergence of the empirical measures, when the size of the system tends to infinity. Then we show the convergence of the fluctuations, considered as cadlag processes with values in a weighted Sobolev space, to an Ornstein-Uhlenbeck process, the solution of a generalized Langevin equation. The tightness is proved by using a Hilbertian approach. The uniqueness of the limit is obtained by considering it as the solution of an evolution equation in a greater Banach space. The main difficulties are due to the unboundedness of the operators appearing in the semimartingale decomposition.