Comptes Rendus
no. 7-8
Probability Theory
Weak uniqueness of Fokker–Planck equations with degenerate and bounded coefficients
[Unicité faible des équations de Fokker–Planck avec coefficients dégénérées et bornées]

Présenté par : Paul Malliavin

Michael Röckner1, 2 ; Xicheng Zhang3
1 Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
2 Departments of Mathematics and Statistics, Purdue University, W. Laffayette, IN 47907, USA
3 Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China
Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 435-438.

Dans cette Note, en utilisant la théorie des équations différentielles stochastiques (EDS), nous démontrons l'unicité de solutions Lp et à valeurs mesures pour des équations de Fokker–Planck du second ordre dégénérées, sous des conditions faibles sur les coefficients. Nos résultats d'unicité sont fondés sur le lien naturel existant entre les équations de Fokker–Planck et les EDS.

In this Note, by using the theory of stochastic differential equations (SDE), we prove uniqueness of measure-valued solutions and Lp-solutions to degenerate second order Fokker–Planck equations under weak conditions on the coefficients. Our uniqueness results are based on the natural connection between Fokker–Planck equations and SDEs.

Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.01.001

Michael Röckner 1, 2 ; Xicheng Zhang 3

1 Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
2 Departments of Mathematics and Statistics, Purdue University, W. Laffayette, IN 47907, USA
3 Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China 
@article{CRMATH_2010__348_7-8_435_0,
     author = {Michael R\"ockner and Xicheng Zhang},
     title = {Weak uniqueness of {Fokker{\textendash}Planck} equations with degenerate and bounded coefficients},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {435--438},
     publisher = {Elsevier},
     volume = {348},
     number = {7-8},
     year = {2010},
     doi = {10.1016/j.crma.2010.01.001},
     language = {en},
}
TY  - JOUR
AU  - Michael Röckner
AU  - Xicheng Zhang
TI  - Weak uniqueness of Fokker–Planck equations with degenerate and bounded coefficients
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 435
EP  - 438
VL  - 348
IS  - 7-8
PB  - Elsevier
DO  - 10.1016/j.crma.2010.01.001
LA  - en
ID  - CRMATH_2010__348_7-8_435_0
ER  - 
%0 Journal Article
%A Michael Röckner
%A Xicheng Zhang
%T Weak uniqueness of Fokker–Planck equations with degenerate and bounded coefficients
%J Comptes Rendus. Mathématique
%D 2010
%P 435-438
%V 348
%N 7-8
%I Elsevier
%R 10.1016/j.crma.2010.01.001
%G en
%F CRMATH_2010__348_7-8_435_0
Michael Röckner; Xicheng Zhang. Weak uniqueness of Fokker–Planck equations with degenerate and bounded coefficients. Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 435-438. doi : 10.1016/j.crma.2010.01.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.001/

[1] V.I. Bogachev; G. Da Prato; M. Röckner; W. Stannat Uniqueness of solutions to weak parabolic equations for measures, Bull. Lond. Math. Soc., Volume 39 (2007) no. 4, pp. 631-640

[2] G. Crippa; C. De Lellis Estimates and regularity results for the DiPerna–Lions flow, J. Reine Angew. Math., Volume 616 (2008), pp. 15-46

[3] L.C. Evans; R.F. Gariepy Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, London, 1992

[4] A. Figalli Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients, J. Funct. Anal., Volume 254 (2008) no. 1, pp. 109-153

[5] N. Ikeda; S. Watanabe Stochastic Differential Equations and Diffusion Processes, North-Holland/Kodanska, Amsterdam/Tokyo, 1981

[6] C. Le Bris; P.L. Lions Existence and uniqueness of solutions to Fokker–Planck type equations with irregular coefficients, Comm. in Partial Differential Equations, Volume 33 (2008), pp. 1272-1317

[7] E.M. Stein Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970

[8] X. Zhang, Stochastic flows of SDEs with irregular coefficients and stochastic transport equations, Bull. Sci. Math. France, , in press | DOI

Cité par Sources :

Commentaires - Politique