STRONG EXISTENCE AND UNIQUENESS FOR STOCHASTIC DIFFERENTIAL EQUATION WITH HÖLDER DRIFT AND DEGENERATE NOISE

Abstract. In this paper, we prove pathwise uniqueness for stochastic degenerate systems with a Hölder drift, for a Hölder exponent larger than the critical value 2/3. This work extends to the degenerate setting the earlier results obtained by Zvonkin [Zvo74], Veretennikov [Ver80], Krylov and Röckner [KR05] from non-degenerate to degenerate cases. The existence of a threshold for the Hölder 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.