Non-equilibrium fluctuations for the stirring process with births and deaths

We consider the one-dimensional stirring process on the segment $\{-N,\dots ,N\}$, coupled to boundary dynamics that inject particles from the right reservoir and remove particles from the left reservoir, each acting on a window of size $K$. We investigate the non-equilibrium fluctuations of the system, starting from a product measure associated with a smooth initial profile. Given our initial state, the fluctuations are given by an Ornstein-Uhlenbeck process whose characteristic operators are the Laplacian and gradient operators. The domains of these operators include functions with boundary conditions that depend on the hydrodynamic profile. A central ingredient in our analysis is the derivation of sharp bounds on the space and space-time $v$-functions of arbitrary degree for the centered occupation variables. In particular, we prove that the $v$-functions of degree $2$ and $3$ are of order $N^{-1}$, while those of degree at least $4$ are of order $N^{-1-\zeta }$ for some $\zeta >0$.