PUB - Publikationen an der Universität Bielefeld

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.