Strong existence and uniqueness for stochastic differential equation with H{\"o}lder drift and degenerate noise

TL;DR

This paper establishes pathwise uniqueness for stochastic differential equations with degenerate noise and Hölder continuous drift, extending classical results to degenerate cases using PDE regularization techniques.

Contribution

It extends existing non-degenerate SDE results to degenerate cases, identifying a Hölder exponent threshold for uniqueness and employing a PDE-based proof via the parametrix method.

Findings

Pathwise uniqueness holds for Hölder exponent > 2/3 in degenerate SDEs.

The work generalizes classical results to systems with degenerate noise.

A PDE regularization approach is used to prove the main results.

Abstract

In this paper, we prove pathwise uniqueness for stochastic degenerate systems with a H{\"o}lder drift, for a H{\"o}lder exponent larger than the critical value 2/3. This work extends to the degenerate setting the earlier results obtained by Zvonkin, Veretennikov, Krylov and R{\"o}ckner from non-degenerate to degenerate cases. The existence of a threshold for the H{\"o}lder exponent in the degenerate case may be understood as the price to pay to balance the degeneracy of the noise. Our proof relies on regularization properties of the associated PDE, which is degenerate in the current framework and is based on a parametrix method.