General Extinction Results for Stochastic Partial Differential Equations and Applications

TL;DR

This paper establishes general extinction results for stochastic partial differential equations using Itô inequalities and applies these findings to fast diffusion equations with fractional Laplacians, including models for self-organized criticality.

Contribution

It introduces a unified approach to prove extinction in stochastic PDEs and applies it to fractional Laplacian diffusion and Zhang-model for criticality.

Findings

Extinction occurs in finite time for a broad class of stochastic PDEs.

Exponential integrability of extinction time is proven for the Zhang-model.

Systems reach critical states in finite time, both deterministically and stochastically.

Abstract

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.