Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions

Abstract

If X = X(t, ξ) is the solution to the stochastic porous media equation in ${\mathcal{O}\subset \mathbf{R}^d, 1\le d\le 3,}$ modelling the self-organized criticality (Barbu et al. in Commun Math Phys 285:901–923, 2009) and X _c is the critical state, then it is proved that ${\int^{\infty}_0m(\mathcal{O}{\setminus}\mathcal{O}^t_0)dt<{\infty},\mathbb{P}\hbox{-a.s.}}$ and ${\lim_{t\to{\infty}} \int_\mathcal{O}|X(t)-X_c|d\xi=\ell<{\infty},\ \mathbb{P}\hbox{-a.s.}}$ Here, m is the Lebesgue measure and ${\mathcal{O}^t_c}$ is the critical region ${\{\xi\in\mathcal{O}; X(t,\xi)=X_c(\xi)\}}$ and X _c (ξ) ≤ X(0, ξ) a.e. ${\xi\in\mathcal{O}}$. If the stochastic Gaussian perturbation has only finitely many modes (but is still function-valued), ${\lim_{t \to {\infty}} \int_K|X(t)-X_c|d\xi=0}$ exponentially fast for all compact ${K\subset\mathcal{O}}$ with probability one, if the noise is sufficiently strong. We also recover that in the deterministic case ℓ = 0.