A Decomposition Theorem of Varadhan Type for Co-local Forms on Large Scale Interacting Systems

In our previous article with Yukio Kametani, we investigated the geometric structure underlying a large scale interacting system on infinite graphs, via constructing a suitable cohomology theory called uniformly local cohomology, which reflects the geometric property of the microscopic model, using a class of functions called the uniformly local functions. In this article, we introduce the co-local functions on the geometric structure associated to a large scale interacting system. We may similarly define the notion of uniformly local functions for co-local functions. However, contrary to the functions appearing in our previous article, the co-local functions reflect the stochastic property of the model, namely the probability measure on the configuration space. We then prove a decomposition theorem of Varadhan type for closed co-local forms. The space of co-local functions and forms contain the space of $L^2$-functions and forms. In the last section, we state a conjecture concerning the decomposition theorem for the $L^2$-case.