  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 a Euclidean lattice
$(\mathbb{Z}^d,\mathbb{E}^d)$. Such decomposition 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
decomposition 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.
Our result is applicable for generalized exclusion processes, multi-species
exclusion processes, and more general lattice gas models.

Probability