A new approach to the characterization of closed forms in the nongradient method (Stochastic Analysis on Large Scale Interacting Systems)

Abstract

We announce our recent result on the characterization of closed forms on configuration spaces associated to interacting particle systems. In the context of the study of hydrodynamic limit, closed forms on an infinite configuration space in a L2 space are well studied and their characterization theorem plays an essential role if our model is non-gradient. In this article, we report that closed forms in the set of local functions can be characterized by a similar way as L2 functions but its proof is very simple and completely different from that for L2 functions. With this new observation, we also have an alternative proof of the original characterization theorem in the L2 space, which does not require the sharp estimate of the spectral gap, for the class of lattice gases that are reversible under the Bernoulli measures. Moreover, we extend these characterization theorems for the models in a crystal lattice from Zd.