Weak Well Posedness for Hypoelliptic Stochastic Differential Equation with Singular Drift: A Sharp Result

Abstract

In this paper, we prove weak uniqueness of hypoelliptic stochastic differential equation with H{ö}lder drift, with H{ö}lder exponent strictly greater than 1/3. We then extend to a weak framework the previous work [CdR12] where strong uniqueness was proved when the H{ö}lder 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 H{ö}lder 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.

References in corpus (2)

• Degenerate SDE with Hölder-Dini Drift and Non-Lipschitz Noise Coefficient
• Averaging along irregular curves and regularisation of ODEs

Cited by in corpus (2)

• Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli
• Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result