Characterization of shift-invariant closed $L^2$-forms for large scale interacting systems on the Euclidean Lattice.

We rigorously formulate and prove for a relatively general class of interactions "the characterization of shift-invariant closed $L^2$-forms" for a large scale interacting system on a Euclidean lattice $(\mathbb{Z}^d,\mathbb{E}^d)$. Such characterization of closed forms has played an essential role in proving the diffusive scaling limit of nongradient systems. The universal expression in terms of conserved quantities was sought from observations for specific models, but a precise formulation or rigorous proof up until now had been elusive. Our result is based on the universal characterization of shift-invariant closed local forms studied in our previous article (arXiv:2009.04699). In the present article, we show that the same universal structure also appears for $L^2$-forms. The essential assumptions are: (i) the set of states on each vertex is a finite set, (ii) the measure on the configuration space is the product measure, and (iii) there is a certain uniform spectral gap estimate for the mean field version of the interaction. Our result is applicable for generalized exclusion processes, multi-species exclusion processes, and more general lattice gas models.