  We consider the dyadic model with viscosity and additive Gaussian noise as a
simplified version of the stochastic Navier-Stokes equations, with the purpose
of studying uniqueness and emergence of singularities. We prove path-wise
uniqueness and absence of blow-up in the intermediate intensity of the
non-linearity, morally corresponding to the 3D case, and blow-up for stronger
intensity. Moreover, blow-up happens with probability one for regular initial
data.
