Weak regularization by stochastic drift : result and counter example

Paul-Eric Chaudru de Raynal (1)

1 LAMA - Laboratoire de Mathématiques

In 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.

Domaines: Probabilités [math.PR]

Paul-Eric Chaudru de Raynal. Weak regularization by stochastic drift : result and counter example. Discrete and Continuous Dynamical Systems - Series A, 2018, 38 (3), pp.1269-1291. ⟨10.3934/dcds.2018052⟩. ⟨hal-01333206v2⟩