Weak regularization by stochastic drift : result and counter example

International audienceIn this paper, weak uniqueness of hypoelliptic stochastic differential equation with Hölder drift is proved when the Hölder exponent is strictly greater than 1/3. This result then ``extends'' to a weak framework the previous works \cite{chaudru_de_raynal_strong_2017,wang_degenerate_2016,fedrizzi_regularity_2017}, where strong uniqueness was proved when the regularity index of the drift is strictly greater than 2/3. Part of the result is also shown to be almost sharp thanks to a counter example when the Hölder exponent of the degenerate component is just below 1/3. The approach is based on martingale problem formulation of Stroock and Varadhan and so on smoothing properties of the associated PDE which is, in the current setting, degenerate