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

Barbu V, Röckner M (2012)
Communications in Mathematical Physics 311(2): 539-555.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Barbu, Viorel; Röckner, MichaelUniBi
Abstract / Bemerkung
If X = X(t, xi) is the solution to the stochastic porous media equation in O subset of R-d, 1 <= d <= 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 integral(infinity)(0) m(O\O-0(t))dt < infinity, P-a.s. and lim(t -> 8) integral X-O vertical bar(t) - X-c vertical bar d xi = l < infinity, P-a.s. Here, m is the Lebesgue measure and O-c(t) is the critical region {xi is an element of O; X( t, xi) = X-c(xi)} and X-c(xi) <= X(0, xi) a.e. xi is an element of O. If the stochastic Gaussian perturbation has only finitely many modes (but is still function-valued), limt(t ->infinity) integral(K)vertical bar X(t) - X-c vertical bar d xi = 0 exponentially fast for all compact K subset of O with probability one, if the noise is sufficiently strong. We also recover that in the deterministic case l = 0.
Erscheinungsjahr
2012
Zeitschriftentitel
Communications in Mathematical Physics
Band
311
Ausgabe
2
Seite(n)
539-555
ISSN
0010-3616
Page URI
https://pub.uni-bielefeld.de/record/2493118

Zitieren

Barbu V, Röckner M. Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions. Communications in Mathematical Physics. 2012;311(2):539-555.
Barbu, V., & Röckner, M. (2012). Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions. Communications in Mathematical Physics, 311(2), 539-555. doi:10.1007/s00220-012-1429-8
Barbu, V., and Röckner, M. (2012). Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions. Communications in Mathematical Physics 311, 539-555.
Barbu, V., & Röckner, M., 2012. Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions. Communications in Mathematical Physics, 311(2), p 539-555.
V. Barbu and M. Röckner, “Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions”, Communications in Mathematical Physics, vol. 311, 2012, pp. 539-555.
Barbu, V., Röckner, M.: Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions. Communications in Mathematical Physics. 311, 539-555 (2012).
Barbu, Viorel, and Röckner, Michael. “Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions”. Communications in Mathematical Physics 311.2 (2012): 539-555.

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