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 $\mathcal{O}\subset {\mathbb{R}}^d$, $1\le d\le 3,$ 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 ${\mathcal{O}}_c^t$ is the critical region $\{\xi \in \mathcal{O};$ $X(t,\xi )=X_c(\xi )\}$ and $X_c(\xi )\le X(0,\xi )$ a.e. $\xi \in \mathcal{O}$. 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 $K\subset \mathcal{O}$ with probability one, if the noise is sufficiently strong. We also recover that in the deterministic case $\ell =0$.

References in corpus (1)

• Stochastic Porous Media Equation and Self-Organized Criticality

Cited by in corpus (6)

• Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise
• Finite time extinction for stochastic sign fast diffusion and self-organized criticality
• Finite speed of propagation for stochastic porous media equations
• Singular-degenerate multivalued stochastic fast diffusion equations
• Improved regularity for the stochastic fast diffusion equation
• The stochastic porous media equation in ${\mathbb{R}}^d$