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If $X=X(t,\xi)$ is the solution to the stochastic porous media equation in$\cal O\subset\mathbb{R}^d$, $1\le d\le 3,$ modelling the self-organizedcriticaity and $X_c$ is the critical state, then it is proved that$\int^\9_0m(\cal O\setminus\cal O^t_0)dt<\9,$ $\mathbb{P}{-a.s.}$ and$\lim_{t\to\9}\int_{\cal O}|X(t)-X_c|d\xi=\ell<\9,\ \mathbb{P}{-a.s.}$ Here,$m$ is the Lebesgue measure and $\cal O^t_c$ is the critical region$\{\xi\in\cal O;$ $ X(t,\xi)=X_c(\xi)\}$ and $X_c(\xi)\le X(0,\xi)$ a.e.$\xi\in\cal O$. If the stochastic Gaussian perturbation has only finitely manymodes (but is still function-valued), $\lim_{t\to\9}\int_K|X(t)-X_c|d\xi=0$exponentially fast for all compact $K\subset\cal O$ with probability one, ifthe noise is sufficiently strong. We also recover that in the deterministiccase $\ell=0$.

Stochastic porous media equations and self-organized criticality: convergence to the critical state in all dimensions