  Let $L$ be a positive definite self-adjoint operator on the $L^2$-space
associated to a $\si$-finite measure space. Let $H$ be the dual space of the
domain of $L^{1/2}$ w.r.t. $L^2(\mu)$. By using an It\^o type inequality for
the $H$-norm and an integrability condition for the hyperbound of the semigroup
$P_t:=\e^{-Lt}$, general extinction results are derived for a class of
continuous adapted processes on $H$. Main applications include stochastic and
deterministic fast diffusion equations with fractional Laplacians. Furthermore,
we prove exponential integrability of the extinction time for all space
dimensions in the singular diffusion version of the well-known Zhang-model for
self-organized criticality, provided the noise is small enough. Thus we obtain
that the system goes to the critical state in finite time in the deterministic
and with probability one in finite time in the stochastic case.
