Finite Time Extinction for Stochastic Sign Fast Diffusion and Self-Organized Criticality

Abstract

Abstract. We prove finite time extinction for stochastic sign fast diffusion equations driven by linear multiplicative space-time noise, corresponding to the Bak- Tang

Existence and extinction in finite time for Stratonovich gradient noise porous media equations

“…Some asymptotic results for the case of SOC in stochastic porous media equations of the type dX − ∆ψ(X) dt = σ(X) dW, have been provided by V. Barbu, G. Da Prato, and M. Röckner in [11,Ch. 3.8] as well as B. Gess in [18], the latter guaranteeing, under some suitable assumptions, the extinction in finite time of solutions also for d > 1. However, in the case of Stratonovich gradient noise, what happens for d > 1 is still to be proved, up to our knowledge, and it could be the next step to be tackled in future works.…”

Well-Posedness of Nonlinear Diffusion Equations with Nonlinear, Conservative Noise

“…In contrast, in the fast diffusion case m ∈ (0, 1), the path-by-path wellposedness of (1.1) and the related problem of the existence of a corresponding stochastic flow and random dynamical system have proven to be notoriously difficult and have been posed as open questions in several works, e.g. Barbu and Röckner [4,5,6] and the second author [11,13]. In addition, even in the porous medium case m ∈ (1, ∞), for general initial data u 0 ∈ L 2 (U ), path-by-path solutions could only be obtained in a limiting sense in [4,6,11], thus lacking a characterization in terms of (generalized) solutions to (1.1).…”

Singular-Degenerate Multivalued Stochastic Fast Diffusion Equations

“…For bounded initial data x 0 ∈ L ∞ (O) and finite driving noise, that is f k ≡ 0 for all k large enough, the existence of strong solutions (cf. Definition A.1 below) to (1.5) has been proven in [9] by entirely different methods, relying on a transformation of (1.5) into a random PDE. Well-posedness for (1.1) with m = 1 and with general multiplicative noise has been obtained in [11] for the first time, proving well-posedness in terms of variational solutions for regular initial data x 0 ∈ L 2 (O).…”