# Stochastic partial differential equations arising in self-organized criticality

@article{Baas2021StochasticPD, title={Stochastic partial differential equations arising in self-organized criticality}, author={Lubom{\'i}r Ba{\~n}as and Benjamin Gess and Marius Neu{\ss}}, journal={ArXiv}, year={2021}, volume={abs/2104.13336} }

Scaling limits for the weakly driven Zhang and the Bak-Tang-Wiesenfeld (BTW) model for self-organized criticality are considered. It is shown that the weakly driven Zhang model converges to a stochastic PDE with singular-degenerate diffusion. In addition, the deterministic BTW model is proved to converge to a singular-degenerate PDE. Alternatively, the proof of convergence can be understood as a proof of convergence of a finite-difference discretization for singular-degenerate stochastic PDE…

## One Citation

Self-Organised Critical Dynamics as a Key to Fundamental Features of Complexity in Physical, Biological, and Social Networks

- PhysicsDynamics
- 2021

Studies of many complex systems have revealed new collective behaviours that emerge through the mechanisms of self-organised critical fluctuations. Subject to the external and endogenous driving…

## References

SHOWING 1-10 OF 90 REFERENCES

Self-organized criticality via stochastic partial differential equations

- Mathematics
- 2008

Models of self-organized criticality, which can be described as singular diffusions with or without (multiplicative) Wiener forcing term (as e.g. the Bak/Tang/Wiesenfeld- and Zhang-models), are…

Singular-Degenerate Multivalued Stochastic Fast Diffusion Equations

- MathematicsSIAM J. Math. Anal.
- 2015

A well-posedness framework based on Stochastic variational inequalities (SVI) is developed, characterizing solutions to the stochastic sign fast diffusion equation, previously obtained in a limiting sense only.

Well-posedness of SVI solutions to singular-degenerate stochastic porous media equations arising in self-organized criticality

- MathematicsStochastics and Dynamics
- 2020

We consider a class of generalized stochastic porous media equations with multiplicative Lipschitz continuous noise. These equations can be related to physical models exhibiting self-organized…

Self-organized criticality and convergence to equilibrium of solutions to nonlinear diffusion equations

- MathematicsAnnu. Rev. Control.
- 2010

Stochastic Porous Media Equations and Self-Organized Criticality

- Mathematics
- 2009

The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time…

Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise

- MathematicsArchive for Rational Mechanics and Analysis
- 2021

We prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal…

Existence of Positive Solutions to Stochastic Thin-Film Equations

- MathematicsSIAM J. Math. Anal.
- 2018

Having established Holder regularity of approximate solutions, the convergence proof is based on compactness arguments---in particular on Jakubowski's generalization of Skorokhod's theorem---weak convergence methods, and recent tools on martingale convergence.