Finite time extinction for stochastic sign fast diffusion and self-organized criticality

Abstract

We prove finite time extinction for stochastic sign fast diffusion equations driven by linear multiplicative space-time noise, corresponding to the Bak-Tang-Wiesenfeld model for self-organized criticality. This solves a problem posed and left open in several works: [Barbu, Da Prato, R\"ockner, CMP, 2009], [Barbu, Da Prato, R\"ockner, CRMAS, 2009], [Barbu, R\"ockner, CMP, 2012], [Barbu, Da Prato, R\"ockner, JMAA, 2012], [R\"ockner, Wang, JMLS, 2013], [Barbu, MMAS, 2013]. The highly singular-degenerate nature of the drift in interplay with the stochastic perturbation causes the need for new methods in the analysis of mass diffusion and several new estimates and techniques are introduced.