# Self-organized criticality via stochastic partial differential equations

@article{Barbu2008SelforganizedCV, title={Self-organized criticality via stochastic partial differential equations}, author={Viorel Barbu and Philippe Blanchard and Giuseppe Da Prato and Michael Rockner}, journal={arXiv: Probability}, year={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 analyzed. Existence and uniqueness of nonnegative strong solutions are proved. Previously numerically predicted transition to the critical state in 1-D is confirmed by a rigorous proof that this indeed happens in finite time with high probability.

## 5 Citations

Stochastic partial differential equations arising in self-organized criticality

- MathematicsArXiv
- 2021

It is shown that the weakly driven Zhang model converges to a stochastic PDE with singular-degenerate diffusion and the deterministic BTW model is proved to converge to a singular- Degenerate PDE.

Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case

- Mathematics
- 2011

We consider a possibly degenerate porous media type equation over all of $${\mathbb R^d}$$ with d = 1, with monotone discontinuous coefficients with linear growth and prove a probabilistic…

Numerical approximation of singular-degenerate parabolic stochastic PDEs

- MathematicsArXiv
- 2020

A fully discrete numerical approximation of the considered SPDEs based on the very weak formulation is proposed and it is proved the convergence of the numerical approximation towards the unique solution is convergence.

Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation

- Mathematics, Computer Science
- 2013

A multi-dimensional generalized porous media equation (PDE) with not smooth and possibly discontinuous coefficient $$\beta $$, which is well-posed as an evolution problem in $$L^1(\mathbb R ^d)$$, which continues the study related to the one-dimensional case.

Stochastic Equations in Infinite Dimensions

- Mathematics
- 2008

Preface Introduction Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. Stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with…

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