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We rigorously formulate and prove for a relatively general class ofinteractions Varadhan's Decomposition of shift-invariant closed $L^2$-forms fora large scale interacting system on a Euclidean lattice$(\mathbb{Z}^d,\mathbb{E}^d)$. Such decomposition of closed forms has played anessential role in proving the diffusive scaling limit of nongradient systems.The universal expression in terms of conserved quantities was sought fromobservations for specific models, but a precise formulation or rigorous proofup until now had been elusive. Our result is based on the universaldecomposition of shift-invariant closed uniform forms studied in our previousarticle (arXiv:2009.04699). In the present article, we show that the sameuniversal structure also appears for $L^2$-forms. The essential assumptionsare: (i) the set of states on each vertex is a finite set, (ii) the measure onthe configuration space is the product measure, and (iii) there is a certainuniform spectral gap estimate for the mean field version of the interaction.Our result is applicable for generalized exclusion processes, multi-speciesexclusion processes, and more general lattice gas models.

Varadhan's Decomposition of Shift-Invariant Closed $L^2$-forms for Large Scale Interacting Systems on the Euclidean Lattice