2021/05/13 by Kenichi Bannai, Bannai, Kenichi, Makiko Sasada +1
Computer Science · Mathematics · #55N91 #60J60 #70G40 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Primary: 82C22 #Probability (math.PR) #Secondary: 05C63 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2105.06043
openalex publication_date 2021/05/13 · openalex created_date 2021/05/24 · openalex updated_date 2026/07/28
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 L2-functions and forms. In the last section, we state a conjecture concerning the decomposition theorem for the L2-case.