Stochastic differential equations with Sobolev drifts and driven by $\alpha$-stable processes

Zhang, Xicheng

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1057-1079.

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

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

[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

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

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

[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

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

[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

[7] Z. Chen, P. Kim and R. Song. Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation. Available at http://arxiv.org/abs/1011.3273.

[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

[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

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

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

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

[13] N. Fournier. On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes. Available at http://arxiv.org/abs/1011.0532. | Numdam | MR

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

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

[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

[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

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

[19] V. P. Kurenok. A note on $L_2$-estimates for stable integrals with drift. Trans. Amer. Math. Soc. 360 (2008) 925-938. | MR

[20] Z. Li and L. Mytnik. Strong solutions for stochastic differential equations with jumps. Available at http://arxiv.org/abs/0910.0950. | MR

[21] E. Priola. Pathwise uniqueness for singular SDEs driven by stable processes. Available at http://arxiv.org/abs/1005.4237. | MR

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

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

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

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

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

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

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

[29] X. Zhang. Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients. Available at http://arxiv.org/abs/1002.4297. | MR

[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