Stochastic PDEs on graphs as scaling limits of discrete interacting systems

TL;DR

This paper establishes a rigorous connection between stochastic PDEs on graphs and discrete interacting particle systems, specifically biased voter models, through scaling limits and simulation schemes, advancing understanding of complex spatial population dynamics.

Contribution

It introduces a new class of SPDEs on graphs with boundary conditions, derived as scaling limits of biased voter models, and provides a simulation method linking discrete models to continuous SPDEs.

Findings

SPDEs on graphs arise as scaling limits of biased voter models.

A convergent simulation scheme using Itô SDEs is developed.

Provides the first rigorous link between SPDEs on graphs and discrete particle systems.

Abstract

Stochastic partial differential equations (SPDE) on graphs were introduced by Cerrai and Freidlin [Ann. Inst. Henri Poincar\'e 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 It\^o 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.