For divergence-free Hölder drifts on T², the vanishing-noise limit of SDEs with fBm or stable-Lévy perturbations does not select a unique Lagrangian trajectory: even and odd subsequences of the stochastic flow separate on a large set of initial data.
Boutros, Camillo De Lellis, and Svitlana Mayboroda
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Non-selection of Lagrangian trajectories in the zero-noise limit for a class of stochastic regularizations
For divergence-free Hölder drifts on T², the vanishing-noise limit of SDEs with fBm or stable-Lévy perturbations does not select a unique Lagrangian trajectory: even and odd subsequences of the stochastic flow separate on a large set of initial data.