Space-time bounds for the stirring process with births and deaths
We consider a one-dimensional symmetric simple exclusion process on the segment $\{-N,\ldots,N\}$, 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 derive sharp bounds on the space and space-time $v$-functions of arbitrary degree associated with the centered occupation variables. In particular, we establish bounds of order $N^{-1+2ζ}$ for spatial $v$-functions of degrees $1$ and $2$, of order $N^{-1}$ for degrees $3,\ldots,K$, and of order $N^{-1-ζ}$ for degrees at least $K+1$, for some sufficiently small $ζ>0$. For space-time $v$-functions, we obtain bounds of order $N^{-1}$ for total degrees $2$ and $3$, together with improved bounds for higher degrees, incorporating the dependence on the time separation. These estimates provide the quantitative control needed to study the non-equilibrium fluctuations of the model.