Stochastic PDEs on graphs as scaling limits of discrete interacting systems

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 in infinite dimensions provides a minimal framework for the study effective dynamics 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 genealogies expanding populations spatially structured environments, we obtain new SPDE Wright–Fisher type which have nontrivial boundary conditions set. We show that these arise as scaling limits suitably defined biased voter (BVM), extends Durrett Fan Appl. 26 (2016) 3456–3490). further convergent simulation scheme each terms system Itô SDEs, useful when size BVM too large simulations. These give first rigorous connection between discrete models, specifically, interacting particle systems SDEs. Uniform heat kernel estimates symmetric random walks approximating diffusions are keys our proofs. Some open problems provided motivations study.