Weak regularization by stochastic drift : Result and counter example

Paul-Eric Chaudru De Raynal

doi:10.3934/dcds.2018052

Article Contents

2018, Volume 38, Issue 3: 1269-1291. Doi: 10.3934/dcds.2018052

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Weak regularization by stochastic drift : Result and counter example

Paul-Eric Chaudru De Raynal

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

Received: February 28, 2017

Revised: August 31, 2017

Published: June 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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