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Multidimensional Singular Stochastic Differential Equations

  • Xicheng ZhangEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)

Abstract

In this paper we survey recent progress about multidimensional stochastic differential equations with singular drifts and Sobolev diffusion coefficients. Moreover, applications to Navier–Stokes equations and SPDEs are also presented.

Keywords

Stochastic flow Krylov’s estimate Zvonkin’s transformation 

AMS 2010 Mathematics Subject Classification

60H10 60J60 

References

  1. 1.
    Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158(2), 227–260 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bogachev V.I. , Pilipenko A.Y.: Strong solutions to stochastic equations with Lévy noise and a discontinuous coefficient. Doklady Math. 92, English transl. No.1, 471–475 (2015)Google Scholar
  3. 3.
    Bouchut, F.: Hypoeliptic regularity in kinetic equations. J. Math. Pures Appl. 81, 1135–1159 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bramanti, M., Cupini, G., Lanconelli, E., Priola, E.: Global \(L^p\)-estimate for degenerate Ornstein–Uhlenbeck operators. Math Z. 266, 789–816 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chaudru de Raynal P. E.: Strong Existence and Uniqueness for Stochastic Differential Equation with Hölder Drift and Degenerate Noise. Annales de l’Institut Henri Poincaré, 53(1), 259–286 (2017)Google Scholar
  6. 6.
    Chen Z.-Q., Zhang X.: \(L^p\)-Maximal Hypoelliptic Regularity of Nonlocal Kinetic Fokker–Planck Operators. J. Math. Pures Appl. (2017).  https://doi.org/10.1016/j.matpur.2017.10.003
  7. 7.
    Chen Z.-Q., Song R., Zhang X.: Stochastic Flows for Lévy Processes with Hölder Drifts. Rev. Mat. Iberoam (2018)Google Scholar
  8. 8.
    Cherny A.S., Engelbert H.J.: Singular Stochastic Differential Equations. Lecture Notes in Mathematics, vol. 1858. Springer, Berlin (2005)Google Scholar
  9. 9.
    Constantin, P., Iyer, G.: A stochastic Lagrangian representation of the three-dimensional incompressible Navier–Stokes equations. Commun. Pure Appl. Math. 61(3), 330–345 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Crippa, G., De Lellis, C.: Estimates and regularity results for the DiPerna–Lions flow. J. Reine Angew. Math. 616, 15–46 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Da Prato, G., Flandoli, F., Priola, E., Röckner, M.: Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Prob. 41(5), 3306–3344 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Da Prato, G., Flandoli F., Röckner M., Veretennikov A.Yu.: Strong uniqueness for SDEs in Hilbert spaces with nonregular drift. Ann. Prob. 44(3), 1985–2023 (2016)Google Scholar
  13. 13.
    Da Prato, G., Flandoli, F., Priola, E., Röckner, M.: Strong uniqueness for stochastic evolution equations with unbounded measurable drift term. J. Theor. Prob. 28(4), 1571–1600 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fang, S., Luo, D., Thalmaier, A.: Stochastic differential equations with coefficients in Sobolev spaces. J. Funct. Anal. 259(5), 1129–1168 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fedrizzi, E., Flandoli, F.: Noise prevents singularities in linear transport equations. J. Funct. Anal. 264, 1329–1354 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Fedrizzi, E., Flandoli, F.: Hölder flow and differentiability for SDEs with non regular drift. Stoch. Anal. Appl. 31, 708–736 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fedrizzi E., Flandoli F., Priola E., Vovelle J.: Regularity of Stochastic Kinetic Equations. http://arXiv.org/pdf/1606.01088v2.pdf
  19. 19.
    Figalli, A.: Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254(1), 109–153 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Flandoli, F., Gubinelli, M., Priola, E.: Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180(1), 1–53 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland/Kodanska, Amsterdam (1989)zbMATHGoogle Scholar
  22. 22.
    Krylov N.V.: Controlled Diffusion Processes. Translated from the Russian by A.B. Aries. Applications of Mathematics, 14. Springer, New York (1980)Google Scholar
  23. 23.
    Krylov, N.V., Röckner, M.: Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields 131, 154–196 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kim, K.-H.: \(L^q(L^p)\)-theory of parabolic PDEs with variable coefficients. Bull. Korean Math. Soc. 45, 169–190 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Menoukeu-Pamen O, Meyer-Brandis T., Nilssen T., Proske F., Zhang T.: A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s. Math. Ann. 357(2), 761–799 (2013)Google Scholar
  26. 26.
    Mohammed, S.E.A., Nilssen, T., Proske, F.: Sobolev differentiable stochastic flows for SDEs with singular coefficients: applications to the transport equation. Ann. Probab. 43(3), 1535–1576 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Priola, E.: Pathwise uniqueness for singular SDEs driven by stable processes. Osaka J. Math. 49, 421–447 (2012)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Rozovskii B.L.: Stochastic Evolution Systems: Linear Theory and Applications to Nonlinear Filtering. Math. Appl. (Sov. Ser.), vol. 35. Kluwer Academic Publishers, Dordrecht (1990)Google Scholar
  29. 29.
    Soize C.: The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions. Ser. Adv. Math. Appl. Sci., vol. 17, World Scientific, Singapore (1994)Google Scholar
  30. 30.
    Stroock, D., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1997)CrossRefGoogle Scholar
  31. 31.
    Talay, D.: Stochastic Hamiltonian systems: exponential convergence to the invariant measure and discretization by the implicit Euler scheme. Markov Process. Related Fields 8, 1–36 (2002)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Veretennikov, AJu: On the strong solutions of stochastic differential equations. Theory Probab. Appl. 24, 354–366 (1979)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wang, F., Zhang, X.: Degenerate SDE with Hölder-Dini drift and non-Lipschitz noise coefficient. SIAM J. Math. Anal. 48(3), 2189–2226 (2016)Google Scholar
  34. 34.
    Xie, L., Zhang, X.: Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients. Ann. Probab. 44(6), 3661–3687 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Xie L., Zhang X.: Ergodicity of Stochastic Differential Equations with Jumps and Singular Coefficients. PreprintGoogle Scholar
  36. 36.
    Zhang, X.: Strong solutions of SDEs with singular drift and Sobolev diffusion coefficients. Stoch. Proc. Appl. 115, 1805–1818 (2005)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Zhang, X.: Stochastic flows of SDEs with irregular coefficients and stochastic transport equations. Bull. Sci. Math. 134(4), 340–378 (2010)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Zhang, X.: Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients. Electron. J. Probab. 16(38), 1096–1116 (2011)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Zhang, X.: Stochastic partial differential equations with unbounded and degenerate coefficients. J. Differ. Equ. 250, 1924–1966 (2011)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Zhang, X.: Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients. Rev. Mat. Iberoam. 29(1), 25–52 (2013)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Zhang X.: Degenerate irregular SDEs with jumps and application to integro-differential equations of Fokker-Planck type. Electron. J. Probab. 18, Article 55, 1–25 (2013)Google Scholar
  42. 42.
    Zhang, X.: Stochastic differential equations with Sobolev drifts and driven by \(\alpha \)-stable processes. Ann. Inst. H. Poincare Probab. Stat. 49, 1057–1079 (2013)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Zhang, X.: Stochastic differential equations with Sobolev diffusion and singular drift. Ann. Appl. Probab. 26(5), 2697–2732 (2016)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Zhang X.: Stochastic Hamiltonian Flows with Singular Coefficients. Sci. China Math. http://engine.scichina.com/doi/10.1007/s11425-017-9127-0
  45. 45.
    Zvonkin A.K.: A Transformation of the Phase Space of a Diffusion Process that Removes the Drift. Mat. Sbornik, 93(135), No. 1, 129-149 (1974)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhan, HubeiChina

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