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

Abstract

We rigorously formulate and prove for a relatively general class of interactions Varadhan's Decomposition of shift-invariant closed $L^2$-forms for a large scale interacting system on the Euclidean lattice with finite range. Such decomposition of closed forms has played an essential role in proving the diffusive scaling limit of nongradient systems. A general 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 a general decomposition theorem of shift-invariant closed uniform 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. As a special case, our result gives Varadhan's decomposition for the case of the multi-species exclusion process, which is a new result that could not be proved by existing methods. Our result also gives complete proofs of Varadhan's decompositions for finite range interactions, whose detailed proofs have been missing in the literature - presumably because there exists an obstruction to the standard method of proof.