WEAK WELL POSEDNESS FOR HYPOELLIPTIC STOCHASTIC DIFFERENTIAL EQUATION WITH SINGULAR DRIFT: A SHARP RESULT

Paul-Eric Chaudru de Raynal (1)

1 LAMA - Laboratoire de Mathématiques

In this paper, we prove weak uniqueness of hypoelliptic stochastic differential equation with Hölder drift, with Hölder exponent strictly greater than 1/3. We then extend to a weak framework the previous work [CdR12] where strong uniqueness was proved when the Hölder exponent is strictly greater than 2/3. We also show that this result is sharp, by giving a counter example to weak uniqueness when the Hölder exponent is just below 1/3. Our approach is based on martingale problem formulation of Stroock and Varadhan and is based on smoothing properties of the associated PDE.

Domaines: Probabilités [math.PR]

Paul-Eric Chaudru de Raynal. WEAK WELL POSEDNESS FOR HYPOELLIPTIC STOCHASTIC DIFFERENTIAL EQUATION WITH SINGULAR DRIFT: A SHARP RESULT. 2016. ⟨hal-01333206v1⟩