Abstract
If X = X(t, ξ) is the solution to the stochastic porous media equation in \({\mathcal{O}\subset \mathbf{R}^d, 1\le d\le 3,}\) 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 \({\int^{\infty}_0m(\mathcal{O}{\setminus}\mathcal{O}^t_0)dt<{\infty},\mathbb{P}\hbox{-a.s.}}\) and \({\lim_{t\to{\infty}} \int_\mathcal{O}|X(t)-X_c|d\xi=\ell<{\infty},\ \mathbb{P}\hbox{-a.s.}}\) Here, m is the Lebesgue measure and \({\mathcal{O}^t_c}\) is the critical region \({\{\xi\in\mathcal{O}; X(t,\xi)=X_c(\xi)\}}\) and X c (ξ) ≤ X(0, ξ) a.e. \({\xi\in\mathcal{O}}\). If the stochastic Gaussian perturbation has only finitely many modes (but is still function-valued), \({\lim_{t \to {\infty}} \int_K|X(t)-X_c|d\xi=0}\) 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 ℓ = 0.
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Communicated by P. Constantin
Supported by the DFG through SFB 701 and IRTG 1132 as well as by the BiBoS Research Centre. V. Barbu was supported by the project PN-II-PCE-2011-3-0027 financed by UEFISCDI (Romania).
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Barbu, V., Röckner, M. Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions. Commun. Math. Phys. 311, 539–555 (2012). https://doi.org/10.1007/s00220-012-1429-8
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DOI: https://doi.org/10.1007/s00220-012-1429-8