# American Institute of Mathematical Sciences

March  2018, 38(3): 1269-1291. doi: 10.3934/dcds.2018052

## Weak regularization by stochastic drift : Result and counter example

 Univ. Savoie Mont Blanc, CNRS, LAMA, F-73000 Chambéry, France

Received  March 2017 Revised  September 2017 Published  December 2017

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 [4,23,10], 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.

Citation: Paul-Eric Chaudru De Raynal. Weak regularization by stochastic drift : Result and counter example. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1269-1291. doi: 10.3934/dcds.2018052
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