Electronic Journal of Probability

Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients

Dario Trevisan

Full-text: Open access


We investigate well-posedness for martingale solutions of stochastic differential equations, under low regularity assumptions on their coefficients, widely extending the results first obtained by A. Figalli in [19]. Our main results are: a very general equivalence between different descriptions for multidimensional diffusion processes, such as Fokker-Planck equations and martingale problems, under minimal regularity and integrability assumptions; and new existence and uniqueness results for diffusions having weakly differentiable coefficients, by means of energy estimates and commutator inequalities. Our approach relies upon techniques recently developed jointly with L. Ambrosio in [6], to address well-posedness for ordinary differential equations in metric measure spaces: in particular, we employ in a systematic way new representations for commutators between smoothing operators and diffusion generators.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 22, 41 pp.

Received: 29 July 2015
Accepted: 3 March 2016
First available in Project Euclid: 18 March 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 35Q84: Fokker-Planck equations

Fokker-Planck equations martingale problem DiPerna-Lions flows

Creative Commons Attribution 4.0 International License.


Trevisan, Dario. Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients. Electron. J. Probab. 21 (2016), paper no. 22, 41 pp. doi:10.1214/16-EJP4453. https://projecteuclid.org/euclid.ejp/1458325000

Export citation


  • [1] Luigi Ambrosio, Transport equation and cauchy problem for $BV$ vector fields, Invent. Math. 158 (2004), no. 2, 227–260.
  • [2] Luigi Ambrosio and Gianluca Crippa, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, Transport equations and multi-D hyperbolic conservation laws, Lect. Notes Unione Mat. Ital., vol. 5, Springer, Berlin, 2008, pp. 3–57.
  • [3] Luigi Ambrosio and Gianluca Crippa, Continuity equations and ODE flows with non-smooth velocity, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 6, 1191–1244.
  • [4] Luigi Ambrosio and Alessio Figalli, On flows associated to Sobolev vector fields in Wiener spaces: an approach à la DiPerna-Lions, J. Funct. Anal. 256 (2009), no. 1, 179–214.
  • [5] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient flows in metric spaces and in the space of probability measures, second ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.
  • [6] Luigi Ambrosio and Dario Trevisan, Well-posedness of Lagrangian flows and continuity equations in metric measure spaces, Anal. PDE 7 (2014), no. 5, 1179–1234.
  • [7] V. I. Bogachev, G. Da Prato, and M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (2008), no. 1, 397–418.
  • [8] V. I. Bogachev, G. Da Prato, M. Röckner, and W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. Lond. Math. Soc. 39 (2007), no. 4, 631–640.
  • [9] V. I. Bogachev, M. Röckner, and S. V. Shaposhnikov, On uniqueness problems related to the Fokker-Planck-Kolmogorov equation for measures, J. Math. Sci. (N. Y.) 179 (2011), no. 1, 7–47, Problems in mathematical analysis. No. 61.
  • [10] V. I. Bogachev, M. Röckner, and S. V. Shaposhnikov, Uniqueness problems for degenerate Fokker–Planck–Kolmogorov equations, J. Math. Sci. 207 (2015-04-21), no. 2, 147–165.
  • [11] Vladimir I. Bogachev, Michael Röckner, and Stanislav V. Shaposhnikov, On uniqueness of solutions to the cauchy problem for degenerate Fokker–Planck–Kolmogorov equations, J. Evol. Equ. 13 (2013-05-07), no. 3, 577–593.
  • [12] François Bouchut and Gianluca Crippa, Uniqueness, renormalization, and smooth approximations for linear transport equations, SIAM J. Math. Anal. 38 (2006), no. 4, 1316–1328.
  • [13] Nicolas Bouleau and Francis Hirsch, Dirichlet forms and analysis on Wiener space, de Gruyter Studies in Mathematics, vol. 14, Walter de Gruyter & Co., Berlin, 1991.
  • [14] Gianluca Crippa and Camillo De Lellis, Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math. 616 (2008), 15–46.
  • [15] G. Da Prato, F. Flandoli, E. Priola, and M. Röckner, Strong uniqueness for stochastic evolution equations in hilbert spaces perturbed by a bounded measurable drift, Ann. Probab. 41 (2013), no. 5, 3306–3344.
  • [16] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547.
  • [17] Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986, Characterization and convergence.
  • [18] Shizan Fang, Dejun Luo, and Anton Thalmaier, Stochastic differential equations with coefficients in Sobolev spaces, J. Funct. Anal. 259 (2010), no. 5, 1129–1168.
  • [19] Alessio Figalli, Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients, J. Funct. Anal. 254 (2008), no. 1, 109–153.
  • [20] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.
  • [21] N. V. Krylov, On Kolmogorov’s equations for finite-dimensional diffusions, Stochastic PDE’s and Kolmogorov equations in infinite dimensions (Cetraro, 1998), Lecture Notes in Math., vol. 1715, Springer, Berlin, 1999, pp. 1–63.
  • [22] N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Related Fields 131 (2005), no. 2, 154–196.
  • [23] Thomas G. Kurtz and Richard H. Stockbridge, Existence of markov controls and characterization of optimal markov controls, SIAM J. Control Optim. 36 (1998), no. 2, 609–653 (electronic).
  • [24] C. Le Bris and P.-L. Lions, Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients, Comm. Partial Differential Equations 33 (2008), no. 7, 1272–1317.
  • [25] E. Lenglart, D. Lépingle, and M. Pratelli, Présentation unifiée de certaines inégalités de la théorie des martingales, Seminar on Probability, XIV (Paris, 1978/1979) (French), Lecture Notes in Math., vol. 784, Springer, Berlin, 1980, With an appendix by Lenglart, pp. 26–52.
  • [26] De Jun Luo, Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 2, 303–314.
  • [27] Emanuele Paolini and Eugene Stepanov, Decomposition of acyclic normal currents in a metric space, J. Funct. Anal. 263 (2012), no. 11, 3358–3390.
  • [28] Michael Röckner and Xicheng Zhang, Weak uniqueness of Fokker-Planck equations with degenerate and bounded coefficients, C. R. Math. Acad. Sci. Paris 348 (2010), no. 7, 435–438.
  • [29] R. E. Showalter, Monotone operators in banach space and nonlinear partial differential equations, Mathematical Surveys and Monographs, vol. 49, American Mathematical Society, Providence, RI, 1997.
  • [30] S. K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz 5 (1993), no. 4, 206–238.
  • [31] Daniel W. Stroock and S. R. Srinivasa Varadhan, Multidimensional diffusion processes, Classics in Mathematics, Springer-Verlag, Berlin, 2006, Reprint of the 1997 edition.
  • [32] A. Ju. Veretennikov, Strong solutions and explicit formulas for solutions of stochastic integral equations, Mat. Sb. (N.S.) 111(153) (1980), no. 3, 434–452, 480.
  • [33] M. Röckner V.I. Bogachev, N.V. Krylov and S.V. Shaposhnikov, Fokker-Planck-Kolmogorov equations, in preparation.
  • [34] Toshio Yamada and Shinzo Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), 155–167.
  • [35] Xicheng Zhang, Stochastic flows of SDEs with irregular coefficients and stochastic transport equations, Bull. Sci. Math. 134 (2010), no. 4, 340–378.
  • [36] Xicheng Zhang, Stochastic partial differential equations with unbounded and degenerate coefficients, J. Differential Equations 250 (2011), no. 4, 1924–1966.
  • [37] Xicheng Zhang, Degenerate irregular SDEs with jumps and application to integro-differential equations of Fokker-Planck type, Electron. J. Probab. 18 (2013), no. 55, 25.