General Extinction Results for Stochastic Partial Differential Equations and Applications

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.Comment: 19 page

Convergence of operators semigroups generated by elliptic operators

Röckner M, Zhang TS. Convergence of operators semigroups generated by elliptic operators. Osaka Journal of Mathematics. 1997;34(4):923-932

Construction of diffusions on configuration spaces

Ma Z-M, Röckner M. Construction of diffusions on configuration spaces. Osaka Journal of Mathematics. 2000;37(2):273-314

Mehler formula and capacities for infinite-dimensional Ornstein-Uhlenbeck processes with general linear drift

Bogachev VI, Röckner M. Mehler formula and capacities for inifinite dimensional Ornstein-Uhlenbeck processes with General linear drift. Osaka Journal of Mathematics . 1995;32(2):237-274