A diffusion limit for Markov chains with log-linear interaction on a graph
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918139524022272 |
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| author | Puhalskii, Anatolii Shcherbakov, Vadim |
| author_facet | Puhalskii, Anatolii Shcherbakov, Vadim |
| contents | In this paper we establish a diffusion limit for a multivariate continuous time Markov chain whose components are indexed by vertices of a finite graph. The components take values in a common finite set of non-negative integers and evolve subject to a graph based log-linear interaction. We show that if the set of common values of the components expands to the set of all non-negative integers, then a time-scaled and normalised version of the Markov chain converges to a system of interacting Ornstein-Uhlenbeck processes reflected at the origin. This limit is akin to heavy traffic limits in queueing (and our model can be naturally interpreted as a queueing model). Our proof draws on developments in queueing theory and relies on martingale methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A diffusion limit for Markov chains with log-linear interaction on a graph Puhalskii, Anatolii Shcherbakov, Vadim Probability In this paper we establish a diffusion limit for a multivariate continuous time Markov chain whose components are indexed by vertices of a finite graph. The components take values in a common finite set of non-negative integers and evolve subject to a graph based log-linear interaction. We show that if the set of common values of the components expands to the set of all non-negative integers, then a time-scaled and normalised version of the Markov chain converges to a system of interacting Ornstein-Uhlenbeck processes reflected at the origin. This limit is akin to heavy traffic limits in queueing (and our model can be naturally interpreted as a queueing model). Our proof draws on developments in queueing theory and relies on martingale methods. |
| title | A diffusion limit for Markov chains with log-linear interaction on a graph |
| topic | Probability |
| url | https://arxiv.org/abs/2501.09323 |