Stochastic differential equations with Sobolev drifts and driven by α -stable processes

Xicheng Zhang

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 4, page 1057-1079
  • ISSN: 0246-0203

Abstract

top
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 ( 2 d d + 1 , 2 ) . Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

How to cite

top

Zhang, Xicheng. "Stochastic differential equations with Sobolev drifts and driven by $\alpha $-stable processes." Annales de l'I.H.P. Probabilités et statistiques 49.4 (2013): 1057-1079. <http://eudml.org/doc/272007>.

@article{Zhang2013,
abstract = {In this article we prove the pathwise uniqueness for stochastic differential equations in $\mathbb \{R\}^\{d\}$ 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.},
author = {Zhang, Xicheng},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {pathwise uniqueness; symmetric $\alpha $-stable process; Krylov’s estimate; fractional Sobolev space; pathwise uniqueness; symmetric -stable process; Krylov's estimate; fractional Sobolev space},
language = {eng},
number = {4},
pages = {1057-1079},
publisher = {Gauthier-Villars},
title = {Stochastic differential equations with Sobolev drifts and driven by $\alpha $-stable processes},
url = {http://eudml.org/doc/272007},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Zhang, Xicheng
TI - Stochastic differential equations with Sobolev drifts and driven by $\alpha $-stable processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 4
SP - 1057
EP - 1079
AB - In this article we prove the pathwise uniqueness for stochastic differential equations in $\mathbb {R}^{d}$ 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.
LA - eng
KW - pathwise uniqueness; symmetric $\alpha $-stable process; Krylov’s estimate; fractional Sobolev space; pathwise uniqueness; symmetric -stable process; Krylov's estimate; fractional Sobolev space
UR - http://eudml.org/doc/272007
ER -

References

top
  1. [1] H. Airault and X. Zhang. Smoothness of indicator functions of some sets in Wiener spaces. J. Math. Pures Appl.79 (2000) 515–523. Zbl0958.60055MR1759438
  2. [2] D. Aldous. Stopping times and tightness. Ann. Probab.6 (1978) 335–340. Zbl0391.60007MR474446
  3. [3] D. Applebaum. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advance Mathematics 93. Cambridge Univ. Press, Cambridge, UK, 2004. MR2072890
  4. [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. Zbl1039.60056MR1971592
  5. [5] R. Bass. Stochastic differential equations with jumps. Probab. Surv.1 (2004) 1–19. Zbl1189.60114MR2095564
  6. [6] K. Bogdan and T. Jakubowski. Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Comm. Math. Phys.271 (2007) 179–198. Zbl1129.47033MR2283957
  7. [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. Zbl1264.60060
  8. [8] G. Crippa and C. De Lellis. Estimates and regularity results for the DiPerna–Lions flow. J. Reine Angew. Math.616 (2008) 15–46. Zbl1160.34004MR2369485
  9. [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. Zbl1202.60096MR2610386
  10. [10] A. M. Davie. Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. IMRN (2007) Art. ID rnm124. Zbl1139.60028MR2377011
  11. [11] E. Fedrizzi and F. Flandoli. Pathwise uniqueness and continuous dependence for SDEs with nonregular drift. Available at arXiv:1004.3485v1. Zbl1221.60081MR2810591
  12. [12] F. Flandoli, M. Gubinelli and E. Priola. Well-posedness of the transport equation by stochastic perturbation. Invent. Math.180 (2010) 1–53. Zbl1200.35226MR2593276
  13. [13] N. Fournier. On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes. Available at http://arxiv.org/abs/1011.0532. Zbl1273.60069MR3060151
  14. [14] I. Gyöngy and T. Martinez. On stochastic differential equations with locally unbounded drift. Czechoslovak Math. J.51 (2001) 763–783. Zbl1001.60060MR1864041
  15. [15] S. He, J. Wang and J. Yan. Semimartingale Theory and Stochastic Calculus. Science Press and CRC Press, Beijing, 1992. Zbl0781.60002MR1219534
  16. [16] N. V. Krylov. Controlled Diffusion Processes. Applications of Mathematics 14. Springer, New York, Berlin, 1980. Translated from the Russian by A. B. Aries. Zbl0459.93002MR601776
  17. [17] N. V. Krylov and M. Röckner. Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields131 (2005) 154–196. Zbl1072.60050MR2117951
  18. [18] V. P. Kurenok. Stochastic equations with time-dependent drifts driven by Lévy processes. J. Theoret. Probab.20 (2007) 859–869. Zbl1145.60033MR2359059
  19. [19] V. P. Kurenok. A note on L 2 -estimates for stable integrals with drift. Trans. Amer. Math. Soc.360 (2008) 925–938. Zbl1137.60029MR2346477
  20. [20] Z. Li and L. Mytnik. Strong solutions for stochastic differential equations with jumps. Available at http://arxiv.org/abs/0910.0950. Zbl1273.60070MR2884224
  21. [21] E. Priola. Pathwise uniqueness for singular SDEs driven by stable processes. Available at http://arxiv.org/abs/1005.4237. Zbl1254.60063MR2945756
  22. [22] J. Ren and X. Zhang. Limit theorems for stochastic differential equations with discontinuous coefficients. SIAM J. Math. Anal.43 (2011) 302–321. Zbl1227.60077MR2765692
  23. [23] K. I. Sato. Lévy Processes and Infinite Divisible Distributions. Cambridge Univ. Press, Cambridge, 1999. Zbl0973.60001MR1739520
  24. [24] E. M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, NJ, 1970. Zbl0207.13501MR290095
  25. [25] H. Tanaka, M. Tsuchiya and S. Watanabe. Perturbation of drift-type for Lévy processes. J. Math. Kyoto Univ.14 (1974) 73–92. Zbl0281.60064MR368146
  26. [26] H. Triebel. Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, 1978. Zbl0387.46033MR503903
  27. [27] A. J. Veretennikov. On the strong solutions of stochastic differential equations. Theory Probab. Appl.24 (1979) 354–366. Zbl0434.60064MR532447
  28. [28] X. Zhang. Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients. Electron. J. Probab.16 (2011) 1096–1116. Zbl1225.60099MR2820071
  29. [29] X. Zhang. Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients. Available at http://arxiv.org/abs/1002.4297. Zbl1262.60056MR3010120
  30. [30] A. K. Zvonkin. A transformation of the phase space of a diffusion process that removes the drift. Mat. Sb.93 (1974) 129–149. Zbl0306.60049MR336813

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.