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Stabilization and sufficient conditions for mixing by stochastic transportare shown. More precisely, given a second order linear operator with possiblyunstable eigenvalues on a smooth compact Riemannian manifold, it is shown thatthe inclusion of transport noise can imply global asymptotic stability.Moreover, it is shown that an arbitrary large exponential rate of convergencecan be reached, implying enhanced dissipation. The sufficient conditions areshown to be satisfied by the so-called Kraichnan model for stochastic transportof passive scalars in turbulent fluids. In addition, an example is givenshowing that it can be sufficient to force four modes in order to inducestabilization.

Stabilization by transport noise and enhanced dissipation in the Kraichnan model