The total variation flow perturbed by gradient linear multiplicative   noise
  We consider stochastic non-linear diffusion equations with a highly singular diffusivity term and multiplicative gradient-type noise. We study existence and uniqueness of non-negative variational solutions in terms of stochastic variational inequalities. We also show extinction in finite time with probability one. These kind of equations arise, e.g. in the use for simulation of image restoring techniques or for modeling turbulence. 
Michael Rockner, Ionut Munteanu
math.PR math.AP math.FA