2 research outputs found
Stochastic porous media equations and self-organized criticality: convergence to the critical state in all dimensions
If is the solution to the stochastic porous media equation in
, modelling the self-organized
criticaity and is the critical state, then it is proved that
\int^\9_0m(\cal O\setminus\cal O^t_0)dt<\9, and
\lim_{t\to\9}\int_{\cal O}|X(t)-X_c|d\xi=\ell<\9,\ \mathbb{P}{-a.s.} Here,
is the Lebesgue measure and is the critical region
and a.e.
. 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|d\xi=0
exponentially fast for all compact with probability one, if
the noise is sufficiently strong. We also recover that in the deterministic
case