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August 2021 Stochastic PDEs on graphs as scaling limits of discrete interacting systems
Wai-Tong Louis Fan
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Bernoulli 27(3): 1899-1941 (August 2021). DOI: 10.3150/20-BEJ1296


Stochastic partial differential equations (SPDE) on graphs were recently 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) 3456–3490). 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. Some open problems are provided as further motivations of our study.


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Wai-Tong Louis Fan. "Stochastic PDEs on graphs as scaling limits of discrete interacting systems." Bernoulli 27 (3) 1899 - 1941, August 2021.


Received: 1 March 2020; Revised: 1 September 2020; Published: August 2021
First available in Project Euclid: 10 May 2021

Digital Object Identifier: 10.3150/20-BEJ1296

Rights: Copyright © 2021 ISI/BS


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Vol.27 • No. 3 • August 2021
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