Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise

Barbu V, Röckner M. Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise. Archive For Rational Mechanics And Analysis. 2013;209(3):797-834.In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation where is a bounded and open domain in and W(t) is a Wiener process of the form and are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term. One main result established here is that for all initial conditions in , it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality. Thus, one obtains a stochastic version of the (minimal) total variation flow. The new approach developed here also allows us to prove the finite time extinction of solutions in dimensions , which is another main result of this work