Uniqueness of degenerate Fokker-Planck equations with weakly differentiable drift whose gradient is given by a singular integral

Dejun Luo (Chinese Academy of Sciences)


In this paper we prove the uniqueness of solutions to degenerate Fokker-Planck equations with bounded coefficients, under the additional assumptions that the diffusion coefficient has $W^{1,2}_{loc}$ regularity, while the gradient of the drift coefficient is merely given by a singular integral.

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Pages: 1-14

Publication Date: July 12, 2014

DOI: 10.1214/ECP.v19-3407


  • Ambrosio, Luigi. Transport equation and Cauchy problem for $BV$ vector fields. Invent. Math. 158 (2004), no. 2, 227-260. MR2096794
  • Ambrosio, Luigi. Transport equation and Cauchy problem for non-smooth vector fields. Calculus of variations and nonlinear partial differential equations, 1-41, Lecture Notes in Math., 1927, Springer, Berlin, 2008. MR2408257
  • Ambrosio, Luigi; Crippa, Gianluca. Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. Transport equations and multi-D hyperbolic conservation laws, 3-57, Lect. Notes Unione Mat. Ital., 5, Springer, Berlin, 2008. MR2409676
  • Ambrosio, Luigi; Figalli, Alessio. On flows associated to Sobolev vector fields in Wiener spaces: an approach \'a la DiPerna-Lions. J. Funct. Anal. 256 (2009), no. 1, 179-214. MR2475421
  • Ambrosio, L. and Trevisan, D.: Well posedness of Lagrangian flows and continuity equations in metric measure spaces, arXiv:1402.4788
  • Bogachev, V. I.; Da Prato, G.; Röckner, M.; Stannat, W. Uniqueness of solutions to weak parabolic equations for measures. Bull. Lond. Math. Soc. 39 (2007), no. 4, 631-640. MR2346944
  • Bogachev, Vladimir; Da Prato, Giuseppe; Röckner, Michael. Existence and uniqueness of solutions for Fokker-Planck equations on Hilbert spaces. J. Evol. Equ. 10 (2010), no. 3, 487-509. MR2674056
  • Bogachev, Vladimir I.; Röckner, Michael; Shaposhnikov, Stanislav V. On uniqueness of solutions to the Cauchy problem for degenerate Fokker-Planck-Kolmogorov equations. J. Evol. Equ. 13 (2013), no. 3, 577-593. MR3089794
  • Bouchut, François; Crippa, Gianluca. Équations de transport à coefficient dont le gradient est donné par une intégrale singulière. (French) [Transport equations with a coefficient whose gradient is given by a singular integral] Séminaire: Équations aux Dérivées Partielles. 2007–2008, Exp. No. I, 15 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2009. MR2532938
  • Bouchut, François; Crippa, Gianluca. Lagrangian flows for vector fields with gradient given by a singular integral. J. Hyperbolic Differ. Equ. 10 (2013), no. 2, 235-282. MR3078074
  • Champagnat, N. and Jabin, P. E.: Strong solutions to stochastic differential equations with rough coefficients, arXiv:1303.2611
  • Crippa, Gianluca; De Lellis, Camillo. Estimates and regularity results for the DiPerna-Lions flow. J. Reine Angew. Math. 616 (2008), 15-46. MR2369485
  • DiPerna, R. J.; Lions, P.-L. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), no. 3, 511-547. MR1022305
  • Fang, Shizan; Luo, Dejun. Transport equations and quasi-invariant flows on the Wiener space. Bull. Sci. Math. 134 (2010), no. 3, 295-328. MR2607935
  • Fang, Shizan; Luo, Dejun; Thalmaier, Anton. Stochastic differential equations with coefficients in Sobolev spaces. J. Funct. Anal. 259 (2010), no. 5, 1129-1168. MR2652184
  • Figalli, Alessio. Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254 (2008), no. 1, 109-153. MR2375067
  • Ikeda, Nobuyuki; Watanabe, Shinzo. Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. xvi+555 pp. ISBN: 0-444-87378-3 MR1011252
  • Le Bris, C.; Lions, P.-L. Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients. Comm. Partial Differential Equations 33 (2008), no. 7-9, 1272-1317. MR2450159
  • Luo, Dejun. Well-posedness of Fokker-Planck type equations on the Wiener space. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (2010), no. 2, 273-304. MR2669050
  • Luo, De Jun. Fokker-Planck type equations with Sobolev diffusion coefficients and BV drift coefficients. Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 2, 303-314. MR3016531
  • Röckner, Michael; Zhang, Xicheng. Weak uniqueness of Fokker-Planck equations with degenerate and bounded coefficients. C. R. Math. Acad. Sci. Paris 348 (2010), no. 7-8, 435-438. MR2607035
  • Stein, Elias M. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970 xiv+290 pp. MR0290095
  • Zhang, Xicheng. Stochastic flows of SDEs with irregular coefficients and stochastic transport equations. Bull. Sci. Math. 134 (2010), no. 4, 340-378. MR2651896

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