Strong existence and uniqueness for degenerate SDE with Hölder drift

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 ( Mat. Sb. (N.S.) 93(135) (1974) 129–149, 152), Veretennikov ( Mat. Sb. (N.S.) 111(153) (1980) 434–452, 480), Krylov and Röckner ( Probab. Theory Related Fields 131 (2) (2005) 154–196) 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. Résumé. ( L’apparition d’un seuil critique pour l’exposant peut-être vue comme le prix à payer pour la dégénérescence. La preuve repose sur des résultats de régularité de la solution de l’EDP associée, qui est dégénérée, et est basée sur une méthode parametrix.