General extinction results for stochastic partial differential equations and applications

Abstract

Let L be a positive-definite self-adjoint operator on the L2 space associated to a σ-finite measure space. Let H be the dual space of the domain of L1/2 with respect to L2(μ). By using an Itô-type inequality for the H-norm and an integrability condition for the hyperbound of the semigroup Pt≔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.