Stochastic differential equations with Sobolev drifts and driven by α-stable processes
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 4, p. 1057-1079

In this article we prove the pathwise uniqueness for stochastic differential equations in d with time-dependent Sobolev drifts, and driven by symmetric α-stable processes provided that α(1,2) and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when α(2d d+1,2). Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

Dans cet article nous prouvons l’existence et l’unicité d’équations différentielles stochastiques dans d avec terme de dérive dépendant du temps dans un espace de Sobolev et dirigées par un processus de Lévy α-stable symétrique avec α(1,2) et de mesure spectrale non-dégénérée. En particulier, le terme de dérive peut avoir des discontinuités de saut quand α(2d d+1,2). Notre preuve est basée sur des estimations de type Krylov pour des semimartingales purement discontinues.

Classification:  60H10
Keywords: pathwise uniqueness, symmetric α-stable process, Krylov’s estimate, fractional Sobolev space
     author = {Zhang, Xicheng},
     title = {Stochastic differential equations with Sobolev drifts and driven by $\alpha $-stable processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {4},
     year = {2013},
     pages = {1057-1079},
     doi = {10.1214/12-AIHP476},
     zbl = {1279.60074},
     mrnumber = {3127913},
     language = {en},
     url = {}
Zhang, Xicheng. Stochastic differential equations with Sobolev drifts and driven by $\alpha $-stable processes. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 4, pp. 1057-1079. doi : 10.1214/12-AIHP476.

[1] H. Airault and X. Zhang. Smoothness of indicator functions of some sets in Wiener spaces. J. Math. Pures Appl. 79 (2000) 515-523. | MR 1759438

[2] D. Aldous. Stopping times and tightness. Ann. Probab. 6 (1978) 335-340. | MR 474446

[3] D. Applebaum. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advance Mathematics 93. Cambridge Univ. Press, Cambridge, UK, 2004. | MR 2072890

[4] R. Bass. Stochastic differential equations driven by symmetric stable processes. In Seminaire de Probabilities, XXXVI 302-313. Lecture Notes in Math. 1801. Springer, Berlin, 2003. | Numdam | MR 1971592

[5] R. Bass. Stochastic differential equations with jumps. Probab. Surv. 1 (2004) 1-19. | MR 2095564

[6] K. Bogdan and T. Jakubowski. Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Comm. Math. Phys. 271 (2007) 179-198. | MR 2283957

[7] Z. Chen, P. Kim and R. Song. Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation. Available at

[8] G. Crippa and C. De Lellis. Estimates and regularity results for the DiPerna-Lions flow. J. Reine Angew. Math. 616 (2008) 15-46. | MR 2369485

[9] G. Da Prato and F. Flandoli. Pathwise uniqueness for a class of SDE in Hilbert spaces and applications. J. Funct. Anal. 259 (2010) 243-267. | MR 2610386

[10] A. M. Davie. Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. IMRN (2007) Art. ID rnm124. | MR 2377011

[11] E. Fedrizzi and F. Flandoli. Pathwise uniqueness and continuous dependence for SDEs with nonregular drift. Available at arXiv:1004.3485v1. | MR 2810591

[12] F. Flandoli, M. Gubinelli and E. Priola. Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180 (2010) 1-53. | MR 2593276

[13] N. Fournier. On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes. Available at | Numdam | MR 3060151

[14] I. Gyöngy and T. Martinez. On stochastic differential equations with locally unbounded drift. Czechoslovak Math. J. 51 (2001) 763-783. | MR 1864041

[15] S. He, J. Wang and J. Yan. Semimartingale Theory and Stochastic Calculus. Science Press and CRC Press, Beijing, 1992. | MR 1219534

[16] N. V. Krylov. Controlled Diffusion Processes. Applications of Mathematics 14. Springer, New York, Berlin, 1980. Translated from the Russian by A. B. Aries. | MR 601776

[17] N. V. Krylov and M. Röckner. Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 (2005) 154-196. | MR 2117951

[18] V. P. Kurenok. Stochastic equations with time-dependent drifts driven by Lévy processes. J. Theoret. Probab. 20 (2007) 859-869. | MR 2359059

[19] V. P. Kurenok. A note on L 2 -estimates for stable integrals with drift. Trans. Amer. Math. Soc. 360 (2008) 925-938. | MR 2346477

[20] Z. Li and L. Mytnik. Strong solutions for stochastic differential equations with jumps. Available at | MR 2884224

[21] E. Priola. Pathwise uniqueness for singular SDEs driven by stable processes. Available at | MR 2945756

[22] J. Ren and X. Zhang. Limit theorems for stochastic differential equations with discontinuous coefficients. SIAM J. Math. Anal. 43 (2011) 302-321. | MR 2765692

[23] K. I. Sato. Lévy Processes and Infinite Divisible Distributions. Cambridge Univ. Press, Cambridge, 1999. | MR 1739520

[24] E. M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ, 1970. | MR 290095

[25] H. Tanaka, M. Tsuchiya and S. Watanabe. Perturbation of drift-type for Lévy processes. J. Math. Kyoto Univ. 14 (1974) 73-92. | MR 368146

[26] H. Triebel. Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, 1978. | MR 503903

[27] A. J. Veretennikov. On the strong solutions of stochastic differential equations. Theory Probab. Appl. 24 (1979) 354-366. | MR 532447

[28] X. Zhang. Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients. Electron. J. Probab. 16 (2011) 1096-1116. | MR 2820071

[29] X. Zhang. Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients. Available at | MR 3010120

[30] A. K. Zvonkin. A transformation of the phase space of a diffusion process that removes the drift. Mat. Sb. 93 (1974) 129-149. | MR 336813