Higher-Order Gaussian Approximations on Path Space for the Simple Symmetric Exclusion Process

Source abstract

Quantitative Gaussian approximation estimates on path space are established for the nonequilibrium density fluctuations of the simple symmetric exclusion process (SSEP) on a $d$-dimensional discrete torus of mesh size $1/n$, started from a product measure with a smooth density profile. A novel $n$-dependent modified Ornstein-Uhlenbeck process is constructed that incorporates higher-order lattice information while remaining linear and Gaussian. It is shown that, for sufficiently high modification orders, this modification yields quantitative Gaussian approximation error bounds of order $n^{-\min\{d/2,2\}}$ for a class of smooth functionals of the entire fluctuation trajectory. In addition, it is shown that retaining an explicit microscopic correlation correction leaves a remainder of order $n^{-d/2}$, thereby distinguishing contributions to the bound due to correlations from those due to spatial approximation. For nonconstant initial profiles in dimensions $d\ge3$, a separate argument yields matching upper and lower bounds of order $n^{-2}$ for the direct weak error of suitable quadratic test functionals, provided that the modification orders are sufficiently high. The proof is based on a novel path-dependent Trotter formula. Its derivation builds on the introduction of a concept of horizontal absolute continuity for non-anticipative functionals and a corresponding martingale formula for Markov jump processes. This formula is subsequently combined with an integral backward Kolmogorov equation for Gaussian continuation functionals. A second ingredient of the proof is a refinement of the exchangeable-pair argument introduced by the authors in a previous work, thereby improving the central-limit rate $n^{-\min\{d/2,1\}}$ to $n^{-d/2}$ for the initial empirical fluctuations.