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