A Decomposition Theorem of Varadhan Type for Co-local Forms on Large Scale Interacting Systems
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| Format: | Preprint |
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2021
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| _version_ | 1866918346129145856 |
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| author | Bannai, Kenichi Sasada, Makiko |
| author_facet | Bannai, Kenichi Sasada, Makiko |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2105_06043 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A Decomposition Theorem of Varadhan Type for Co-local Forms on Large Scale Interacting Systems Bannai, Kenichi Sasada, Makiko Probability Algebraic Geometry Algebraic Topology Primary: 82C22, Secondary: 05C63, 55N91, 60J60, 70G40 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. |
| title | A Decomposition Theorem of Varadhan Type for Co-local Forms on Large Scale Interacting Systems |
| topic | Probability Algebraic Geometry Algebraic Topology Primary: 82C22, Secondary: 05C63, 55N91, 60J60, 70G40 |
| url | https://arxiv.org/abs/2105.06043 |