On Non-equilibrium Fluctuations for the Stirring Process with Births and Deaths

We study 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 (to be proved in a companion paper [2]) we derive an equation for the limiting kernel of the variance of the fluctuation field. The key difficulty lies in closing the martingales, with respect to the density fluctuation field, due to the boundary effects and we present a strategy on how to deal with more general boundary conditions.