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

Abstract

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-organized criticaity 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 many modes (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, if the noise is sufficiently strong. We also recover that in the deterministic case $\ell=0$.