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 $d=1,2$.

翻译：我们研究张氏和Bak-Tang-Wiesenfeld(BTW)微弱驱动的自我组织临界值模型的缩放限制。 我们发现,微弱驱动的张氏模型与具有单离子扩散的随机偏差方程式(PDE)相融合。 此外, 确定性BTW模型被显示为与单离子PDE相融合。 或者, 缩放限制的证明可以被理解为单离子( 单离子) 随机性PDEs 的有限差异分解的趋同证据。 这将最近关于( 确定性) 准线性扩散方程式的微差近似近度工作延伸至不连续扩散系数和随机PDEs。 此外, 我们还进行数字模拟, 说明所考虑的模型的关键特征以及空间维中与随机PDEs的趋同值 $d=1,2美元。