Stochastic partial differential equations arising in self-organized criticality

TL;DR

This paper investigates the scaling limits of models in self-organized criticality, showing convergence to singular-degenerate stochastic PDEs and providing numerical simulations in 1D and 2D.

Contribution

It establishes the convergence of the weakly driven Zhang and BTW models to singular-degenerate stochastic PDEs, extending finite difference approximation methods to stochastic and discontinuous cases.

Findings

Convergence of Zhang model to a stochastic PDE with singular-degenerate diffusion.

Deterministic BTW model converges to a singular-degenerate PDE.

Numerical simulations confirm the theoretical convergence in 1D and 2D.

Abstract

We study scaling limits of the weakly driven Zhang and the Bak-Tang-Wiesenfeld (BTW) model for self-organized criticality. We show that the weakly driven Zhang model converges to a stochastic partial differential equation (PDE) with singular-degenerate diffusion. In addition, the deterministic BTW model is shown to converge to a singular-degenerate PDE. Alternatively, the proof of the scaling limit can be understood as a convergence proof of a finite-difference discretization for singular-degenerate stochastic PDEs. This extends recent work on finite difference approximation of (deterministic) quasilinear diffusion equations to discontinuous diffusion coefficients and stochastic PDEs. In addition, we perform numerical simulations illustrating key features of the considered models and the convergence to stochastic PDEs in spatial dimension $d=1,2$.