Stochastic PDEs on graphs as scaling limits of discrete interacting systems Article Swipe
Stochastic partial differential equations (SPDE) on graphs were introduced by Cerrai and Freidlin [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 865-899]. This class of stochastic equations in infinite dimensions provides a minimal framework for the study of the effective dynamics of much more complex systems. However, how they emerge from microscopic individual-based models is still poorly understood, partly due to complications near vertex singularities. In this work, motivated by the study of the dynamics and the genealogies of expanding populations in spatially structured environments, we obtain a new class of SPDE on graphs of Wright-Fisher type which have nontrivial boundary conditions on the vertex set. We show that these SPDE arise as scaling limits of suitably defined biased voter models (BVM), which extends the scaling limits of Durrett and Fan [Ann. Appl. Probab. 26 (2016) 456-490]. We further obtain a convergent simulation scheme for each of these SPDE in terms of a system of Itô SDEs, which is useful when the size of the BVM is too large for stochastic simulations. These give the first rigorous connection between SPDE on graphs and more discrete models, specifically, interacting particle systems and interacting SDEs. Uniform heat kernel estimates for symmetric random walks approximating diffusions on graphs are the keys to our proofs.
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- Type
- preprint
- Language
- en
- Landing Page
- https://doi.org/10.3150/20-bej1296
- OA Status
- green
- Cited By
- 2
- References
- 103
- Related Works
- 20
- OpenAlex ID
- https://openalex.org/W2744241661
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https://openalex.org/W2744241661Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.3150/20-bej1296Digital Object Identifier
- Title
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Stochastic PDEs on graphs as scaling limits of discrete interacting systemsWork title
- Type
-
preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2021Year of publication
- Publication date
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2021-05-01Full publication date if available
- Authors
-
Wai-Tong Louis FanList of authors in order
- Landing page
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https://doi.org/10.3150/20-bej1296Publisher landing page
- Open access
-
YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
- OA URL
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https://arxiv.org/pdf/1708.01793Direct OA link when available
- Concepts
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Mathematics, Scaling, Vertex (graph theory), Applied mathematics, Connection (principal bundle), Gravitational singularity, Stochastic differential equation, Random walk, Statistical physics, Pure mathematics, Graph, Mathematical analysis, Discrete mathematics, Physics, Geometry, StatisticsTop concepts (fields/topics) attached by OpenAlex
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2Total citation count in OpenAlex
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2021: 1, 2020: 1Per-year citation counts (last 5 years)
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103Number of works referenced by this work
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20Other works algorithmically related by OpenAlex
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