  In this paper we introduce a probabilistic approach to show the existence of
initial data with arbitrarily large $L^2(\mathbb{R}^3)$,
$\dot{H}^{1/2}(\mathbb{R}^3)$ and $\mathcal{PM}^2$-norms for which a
Generalized Navier-Stokes system generate a global regular solution. More
precisely, we show that from a certain family of possible large initial data
most of them give raise to global regular solutions to a given Generalized
Navier-Stokes system.
