Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Ergodicity of stochastic differential equations with jumps and singular coefficients

Longjie Xie and Xicheng Zhang

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We show the strong well-posedness of SDEs driven by general multiplicative Lévy noises with Sobolev diffusion and jump coefficients and integrable drifts. Moreover, we also study the strong Feller property, irreducibility as well as the exponential ergodicity of the corresponding semigroup when the coefficients are time-independent and singular dissipative. In particular, the large jump is allowed in the equation. To achieve our main results, we present a general approach for treating the SDEs with jumps and singular coefficients so that one just needs to focus on Krylov’s a priori estimates for SDEs.


Nous montrons que les EDS dirigées par un bruit de Lévy multiplicatif général avec des coefficients de diffusion et de saut Sobolev, et une dérive intégrable, sont fortement bien posées. De plus, nous étudions la propriété forte de Feller, l’irréductibilité ainsi que l’ergodicité exponentielle des semi-groupes correspondants quand les coefficients sont indépendants du temps et singulièrement dissipatifs. En particulier, les grands sauts sont autorisés dans l’équation. Pour aboutir au résultat principal, nous présentons une approche générale pour traiter les EDS avec sauts et coefficients singuliers, de telle sorte que nous devons seulement nous intéresser aux estimées a priori de Krylov pour les EDS.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 1 (2020), 175-229.

Received: 11 August 2017
Revised: 4 December 2018
Accepted: 15 January 2019
First available in Project Euclid: 3 February 2020

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Mathematical Reviews number (MathSciNet)

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Pathwise uniqueness Krylov’s estimate Zvonkin’s transformation Ergodicity Heat kernel


Xie, Longjie; Zhang, Xicheng. Ergodicity of stochastic differential equations with jumps and singular coefficients. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 1, 175--229. doi:10.1214/19-AIHP959.

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