In this talk, we present our recent results on the density fluctuation field for the stirring process with births and deaths at two sites at the boundary which tend to impose a current into the system. Assuming correlation estimates, we derive an equation for the limiting kernel of the variance of the fluctuation field. We also present a strategy towards closing the martingales with respect to the density fluctuation field which is the key difficulty of this problem due to the boundary effects. This is a joint work with Dr. D. Tsagkarogiannis and Prof. Patricia Goncalves.