The interaction between noise and transport mechanisms in PDEs

The understanding of transport mechanisms in PDEs is at the core of some of the main open problems in fluid dynamics. The introduction of random perturbations in these equations may have an influence on the properties of transport. We report here some recent progresses on the interaction between noise and transport. The Lagrangian viewpoint is used to investigate the problem. In some example we observe that noise improves the properties of the Lagrangian motion and restores uniqueness or prevents blow-up of the associated PDE; in other examples this does not happen.