Strong regularization by Brownian noise propagating through a weak H{\"o}rmander structure

TL;DR

This paper proves strong uniqueness for certain degenerate stochastic differential equations with H{"o}lder continuous drifts, using a Zvonkin transform and advanced PDE regularization techniques.

Contribution

It introduces new regularization methods for degenerate SDEs with rough coefficients, establishing sharp thresholds for strong uniqueness.

Findings

Established strong uniqueness under H{"o}lder regularity conditions.

Developed a perturbation technique using a forward parametrix approach.

Identified sharp thresholds on H{"o}lder exponents for uniqueness.

Abstract

We establish strong uniqueness for a class of degenerate SDEs of weak H{\"o}rmander type under suitable H{\"o}lder regularity conditions for the associated drift term. Our approach relies on the Zvonkin transform which requires to exhibit good smoothing properties of the underlying parabolic PDE with rough, here H{\"o}lder, drift coefficients and source term. Such regularizing effects are established through a perturbation technique (forward parametrix approach) which also heavily relies on appropriate duality properties on Besov spaces. For the method employed, we exhibit some sharp thresholds on the H{\"o}lder exponents for the strong uniqueness to hold.