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

If X = X(t, ξ) is the solution to the stochastic porous media equation in O ì R^d, 1 ￡ d ￡ 3,{mathcal{O}subset mathbf{R}^d, 1le dle 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 ò^￥₀m(OO^t₀)dt < ￥,mathbbP-a.s.{int^{infty}_0m(mathcal{O}{setminus}mathcal{O}^t_0)dt<{infty},mathbb{P}hbox{-a.s.}} and lim_t?￥ ò_O|X(t)-X_c|dx = l < ￥, mathbbP-a.s.{lim_{tto{infty}} int_mathcal{O}|X(t)-X_c|dxi=ell<{infty}, mathbb{P}hbox{-a.s.}} Here, m is the Lebesgue measure and O^t_c{mathcal{O}^t_c} is the critical region {x ? O; X(t,x)=X_c(x)}{{xiinmathcal{O}; X(t,xi)=X_c(xi)}} and X _c (ξ) ≤ X(0, ξ) a.e. x ? O{xiinmathcal{O}}. If the stochastic Gaussian perturbation has only finitely many modes (but is still function-valued), lim_{t ? ￥} ò_K|X(t)-X_c|dx = 0{lim_{t to {infty}} int_K|X(t)-X_c|dxi=0} exponentially fast for all compact K ì O{Ksubsetmathcal{O}} with probability one, if the noise is sufficiently strong. We also recover that in the deterministic case ℓ = 0.