随机多孔介质方程和自组织临界性：在所有维度上收敛到临界状态
Stochastic porous media equations and self-organized criticality: convergence to the critical state in all dimensions
摘要：
若随机多孔介质方程在$\cal O\subset\mathbb{R}^d$中的解是$X=X(t,ξ)$，$1\le d\le 3,$模拟自组织临界性，$X_c$是临界状态，则证明了$\int^\9_0m(\cal O\setminus\cal O^t_0)dt<\9,$ $\mathbb{P}{-a.s.}$和$\lim_{t\to\9}\int_{\cal O}|X(t)-X_c|dξ=\ell<\9,\ \mathbb{P}{-a.s.}$这里，$m$是Lebesgue测度，$\cal O^t_c$是临界区域$\{ξ\in\cal O;$ $ X(t,ξ)=X_c(ξ)\}$和$X_c(ξ)\le X(0,ξ)$ a.e. $ξ\in\cal O$。如果随机高斯扰动只有100多个模式（但仍然是函数值），$\lim_{t\to\9}\int_K|X(t)-X_c|dξ=0$指数快的概率为1的所有紧凑的$K\subset\cal O$，如果噪音是足够强。我们也恢复，在确定性的情况下$\ell=0$。
英文摘要：
If $X=X(t,ξ)$ 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 $\lim_{t\to\9}\int_{\cal O}|X(t)-X_c|dξ=\ell<\9,\ \mathbb{P}{-a.s.}$ Here, $m$ is the Lebesgue measure and $\cal O^t_c$ is the critical region $\{ξ\in\cal O;$ $ X(t,ξ)=X_c(ξ)\}$ and $X_c(ξ)\le X(0,ξ)$ a.e. $ξ\in\cal O$. If the stochastic Gaussian perturbation has only finitely many modes (but is still function-valued), $\lim_{t\to\9}\int_K|X(t)-X_c|dξ=0$ 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 $\ell=0$.
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