General extinction results for stochastic partial differential equations and applications
Michael Röckner, Feng‐Yu Wang · PUB – Publications at Bielefeld University (Bielefeld University) · 2013
Let L be a positive-definite self-adjoint operator on the L-2 space associated to a sigma-finite measure space. Let H be the dual space of the domain of L-1/2 with respect to L-2(mu). By using an Ito-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 case and with probability 1 in finite time in the stochastic case.