Annals of Probability

Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients

Longjie Xie and Xicheng Zhang

Full-text: Open access

Abstract

By establishing a characterization for Sobolev differentiability of random fields, we prove the weak differentiability of solutions to stochastic differential equations with local Sobolev and super-linear growth coefficients with respect to the starting point. Moreover, we also study the strong Feller property and the irreducibility to the associated diffusion semigroup.

Article information

Source
Ann. Probab., Volume 44, Number 6 (2016), 3661-3687.

Dates
Received: April 2015
Revised: August 2015
First available in Project Euclid: 14 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1479114260

Digital Object Identifier
doi:10.1214/15-AOP1057

Mathematical Reviews number (MathSciNet)
MR3572321

Zentralblatt MATH identifier
06674835

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

Keywords
Weak differentiability Krylov’s estimate Zvonkin’s transformation strong Feller property irreducibility

Citation

Xie, Longjie; Zhang, Xicheng. Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients. Ann. Probab. 44 (2016), no. 6, 3661--3687. doi:10.1214/15-AOP1057. https://projecteuclid.org/euclid.aop/1479114260


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References

  • [1] Adams, R. A. and Fournier, J. J. F. (2003). Sobolev Spaces, 2nd ed. Pure and Applied Mathematics (Amsterdam) 140. Elsevier/Academic Press, Amsterdam.
  • [2] Ambrosio, L. (2004). Transport equation and Cauchy problem for $BV$ vector fields. Invent. Math. 158 227–260.
  • [3] Cerrai, S. (2001). Second Order PDE’s in Finite and Infinite Dimension: A Probabilistic Approach. Lecture Notes in Math. 1762. Springer, Berlin.
  • [4] Champagnat, N. and Jabin, P.-E. (1956). Strong solutions to stochastic differential equations with rough cofficients. Available at http://arxiv.org/abs/1303.2611.
  • [5] Chen, X. and Li, X.-M. (2014). Strong completeness for a class of stochastic differential equations with irregular coefficients. Electron. J. Probab. 19 1–34.
  • [6] Crippa, G. and De Lellis, C. (2008). Estimates and regularity results for the DiPerna–Lions flow. J. Reine Angew. Math. 616 15–46.
  • [7] Da Prato, G. and Flandoli, F. (2010). Pathwise uniqueness for a class of SDE in Hilbert spaces and applications. J. Funct. Anal. 259 243–267.
  • [8] Da Prato, G., Flandoli, F., Priola, E. and Röckner, M. (2013). Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41 3306–3344.
  • [9] Da Prato, G., Flandoli, F., Priola, E. and Röckner, M. (2015). Strong uniqueness for stochastic evolution equations with unbounded measurable drift term. J. Theoret. Probab. 28 1571–1600.
  • [10] Da Prato, G., Flandoli, F., Röckner, M. and Veretennikov, Y. A. Strong uniqueness for SDEs in Hilbert spaces with non-regular drift. Available at arXiv:1404.5418.
  • [11] DiPerna, R. J. and Lions, P.-L. (1989). Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 511–547.
  • [12] Fang, S., Luo, D. and Thalmaier, A. (2010). Stochastic differential equations with coefficients in Sobolev spaces. J. Funct. Anal. 259 1129–1168.
  • [13] Fedrizzi, E. and Flandoli, F. (2011). Pathwise uniqueness and continuous dependence of SDEs with non-regular drift. Stochastics 83 241–257.
  • [14] Fedrizzi, E. and Flandoli, F. (2013). Hölder flow and differentiability for SDEs with nonregular drift. Stoch. Anal. Appl. 31 708–736.
  • [15] Fedrizzi, E. and Flandoli, F. (2013). Noise prevents singularities in linear transport equations. J. Funct. Anal. 264 1329–1354.
  • [16] Flandoli, F., Gubinelli, M. and Priola, E. (2010). Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180 1–53.
  • [17] Gyöngy, I. and Martínez, T. (2001). On stochastic differential equations with locally unbounded drift. Czechoslovak Math. J. 51(126) 763–783.
  • [18] Hajłasz, P. (1996). Sobolev spaces on an arbitrary metric space. Potential Anal. 5 403–415.
  • [19] Krylov, N. V. (1980). Controlled Diffusion Processes. Applications of Mathematics 14. Springer, New York.
  • [20] Krylov, N. V. (1986). Estimates of the maximum of the solution of a parabolic equation and estimates of the distribution of a semimartingale. Mat. Sb. 130(172) 207–221, 284.
  • [21] Krylov, N. V. (2008). Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Graduate Studies in Mathematics 96. Amer. Math. Soc., Providence, RI.
  • [22] Krylov, N. V. and Röckner, M. (2005). Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 154–196.
  • [23] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
  • [24] Li, X.-M. (1994). Strong $p$-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds. Probab. Theory Related Fields 100 485–511.
  • [25] Menoukeu-Pamen, O., Meyer-Brandis, T., Nilssen, T., Proske, F. and Zhang, T. (2013). A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s. Math. Ann. 357 761–799.
  • [26] Mohammed, S.-E. A., Nilssen, T. K. and Proske, F. N. (2015). Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation. Ann. Probab. 43 1535–1576.
  • [27] Prévôt, C. and Röckner, M. (2007). A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Math. 1905. Springer, Berlin.
  • [28] Ren, J., Wu, J. and Zhang, X. (2010). Exponential ergodicity of non-Lipschitz multivalued stochastic differential equations. Bull. Sci. Math. 134 391–404.
  • [29] Rezakhanlou, F. (2014). Regular flows for diffusions with rough drifts. Available at http://arXiv.org/abs/1405.5856.
  • [30] Stein, E. M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30. Princeton Univ. Press, Princeton, NJ.
  • [31] Sznitman, A.-S. (1998). Brownian Motion, Obstacles and Random Media. Springer, Berlin.
  • [32] Veretennikov, A. J. (1979). Strong solutions of stochastic differential equations. Theory Probab. Appl. 24 354–366.
  • [33] Wang, F. Y. and Zhang, X. Degenerate SDE with Hölder-Dini drift and non-Lipschitz noise coefficient. Available at http://arXiv:1504.04450.
  • [34] Zhang, X. Stochastic differential equations with Sobolev coefficients and applications. Available at http://arxiv.org/abs/1406.7446.
  • [35] Zhang, X. (2005). Strong solutions of SDES with singular drift and Sobolev diffusion coefficients. Stochastic Process. Appl. 115 1805–1818.
  • [36] Zhang, X. (2010). Stochastic flows and Bismut formulas for stochastic Hamiltonian systems. Stochastic Process. Appl. 120 1929–1949.
  • [37] Zhang, X. (2011). Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients. Electron. J. Probab. 16 1096–1116.
  • [38] Zhang, X. (2013). Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients. Rev. Mat. Iberoam. 29 25–52.