Electronic Communications in Probability

On differentiability of stochastic flow for а multidimensional SDE with discontinuous drift

Olga Aryasova and Andrey Pilipenko

Full-text: Open access


We consider a d-dimensional SDE with an identity diffusion matrix and a drift vector being a vector function of bounded variation. We give a representation for the derivative of the solution with respect to the initial data.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 45, 17 pp.

Accepted: 15 July 2014
First available in Project Euclid: 7 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Stochastic flow Continuous additive functional Differentiability with respect to initial data

This work is licensed under a Creative Commons Attribution 3.0 License.


Aryasova, Olga; Pilipenko, Andrey. On differentiability of stochastic flow for а multidimensional SDE with discontinuous drift. Electron. Commun. Probab. 19 (2014), paper no. 45, 17 pp. doi:10.1214/ECP.v19-2886. https://projecteuclid.org/euclid.ecp/1465316747

Export citation


  • Aizenman, M.; Simon, B. Brownian motion and Harnack inequality for Schrödinger operators. Comm. Pure Appl. Math. 35 (1982), no. 2, 209–273.
  • Aronson, D. G. Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 1967 890–896.
  • Aryasova, Olga V.; Pilipenko, Andrey Yu. On properties of a flow generated by an SDE with discontinuous drift. Electron. J. Probab. 17 (2012), no. 106, 20 pp.
  • Attanasio, Stefano. Stochastic flows of diffeomorphisms for one-dimensional SDE with discontinuous drift. Electron. Commun. Probab. 15 (2010), 213–226.
  • Bass, Richard F.; Chen, Zhen-Qing. Brownian motion with singular drift. Ann. Probab. 31 (2003), no. 2, 791–817.
  • Blumenthal, R. M.; Getoor, R. K. Markov processes and potential theory. Pure and Applied Mathematics, Vol. 29 Academic Press, New York-London 1968 x+313 pp.
  • Bogachev, V. I. Measure theory. Vol. I, II. Springer-Verlag, Berlin, 2007. Vol. I: xviii+500 pp., Vol. II: xiv+575 pp. ISBN: 978-3-540-34513-8; 3-540-34513-2
  • Dynkin, E. B. Markov processes. Vols. I, II. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bönde 121, 122 Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg 1965 Vol. I: xii+365 pp.; Vol. II: viii+274 pp.
  • Federer, Herbert. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969 xiv+676 pp.
  • Fedrizzi, E.; Flandoli, F. Hölder Flow and Differentiability for SDEs with Nonregular Drift. Stoch. Anal. Appl. 31 (2013), no. 4, 708–736.
  • Fedrizzi, E.; Flandoli, F. Noise prevents singularities in linear transport equations. J. Funct. Anal. 264 (2013), no. 6, 1329–1354.
  • Flandoli, F.; Gubinelli, M.; Priola, E. Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift. Bull. Sci. Math. 134 (2010), no. 4, 405–422.
  • Gikhman, Iosif I.; Skorokhod, Anatoli V. The theory of stochastic processes. II. Translated from the Russian by S. Kotz. Reprint of the 1975 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2004. viii+441 pp. ISBN: 3-540-20285-4
  • HasʹminskiÄ­, R. Z. On positive solutions of the equation ${\cal U}+Vu=0$. Theor. Probability Appl. 4 1959 309–318.
  • Kulik, A. M.; Pilipenko, A. Yu. Nonlinear transformations of smooth measures on infinite-dimensional spaces. (Russian) Ukrain. Mat. Zh. 52 (2000), no. 9, 1226–1250; translation in Ukrainian Math. J. 52 (2000), no. 9, 1403–1431 (2001)
  • Kuwae, Kazuhiro; Takahashi, Masayuki. Kato class measures of symmetric Markov processes under heat kernel estimates. J. Funct. Anal. 250 (2007), no. 1, 86–113.
  • Liptser, R. S.; Shiryayev, A. N. Statistics of random processes. I. General theory. Translated by A. B. Aries. Applications of Mathematics, Vol. 5. Springer-Verlag, New York-Heidelberg, 1977. x+394 pp. ISBN: 0-387-90226-0
  • Luo, Dejun. Absolute continuity under flows generated by SDE with measurable drift coefficients. Stochastic Process. Appl. 121 (2011), no. 10, 2393–2415.
  • Meyer-Brandis, Thilo; Proske, Frank. Construction of strong solutions of SDE's via Malliavin calculus. J. Funct. Anal. 258 (2010), no. 11, 3922–3953.
  • S.E.A. Mohammed, T. Nilssen, and F. Proske. Sobolev differentiable stochastic flows of SDE's with measurable drift and applications. 2012. arXiv/1204.3867.
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7
  • Sznitman, Alain-Sol. Brownian motion, obstacles and random media. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xvi+353 pp. ISBN: 3-540-64554-3
  • Veretennikov, A. Ju. Strong solutions and explicit formulas for solutions of stochastic integral equations. (Russian) Mat. Sb. (N.S.) 111(153) (1980), no. 3, 434–452, 480.
  • V. S. Vladimirov. The Equation of Mathematical Phisics. Nauka, Moscow, 1967. [Translated from the Russian to the English by A. Littlewood. Marcel Dekker, INC., New York, 1971.].