Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Strong existence and uniqueness for degenerate SDE with Hölder drift

P. E. Chaudru de Raynal

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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

Dans ce travail, on montre qu’un système hypoelliptique, composé d’une composante diffusive et d’une composante totalement dégénérée, est fortement résoluble lorsque l’exposant de la régularité Hölder de la dérive par rapport à la composante dégénérée est strictement supérieur à $2/3$. Ce travail étend au cadre dégénéré les travaux antérieurs de Zvonkin (Mat. Sb. (N.S.) 93(135) (1974) 129–149, 152), Veretennikov (Mat. Sb. (N.S.) 111(153) (1980) 434–452, 480), Krylov et Röckner (Probab. Theory Related Fields 131(2) (2005) 154–196). 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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 1 (2017), 259-286.

Dates
Received: 5 June 2012
Revised: 14 September 2015
Accepted: 14 September 2015
First available in Project Euclid: 8 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1486544892

Digital Object Identifier
doi:10.1214/15-AIHP716

Mathematical Reviews number (MathSciNet)
MR3606742

Zentralblatt MATH identifier
06701701

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.) 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]

Keywords
Strong uniqueness Degeneracy Hölder drift Parametrix Stochastic differential equation

Citation

Chaudru de Raynal, P. E. Strong existence and uniqueness for degenerate SDE with Hölder drift. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 1, 259--286. doi:10.1214/15-AIHP716. https://projecteuclid.org/euclid.aihp/1486544892


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