Finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise

In this paper one surveys and sharpens some recent results obtained by the authors on the finite time extinction property of solutions to stochastic diffusion equations of the form d X − ρ Δ X m d t = X d W and d X − ρ Δ sgn X d t = X d W , where 0 < m < 1 , ρ > 0 . These equations arise as models for nonlinear diffusion processes in porous media, plasma and self-organized criticality under stochastic Gaussian perturbation. These equations can be also viewed as control systems governed by fast diffusion porous media equations with a stochastic feedback controller X d W .