Degenerate irregular SDEs with jumps and application to integro-differential equations of Fokker-Planck type

Xicheng Zhang (Wuhan University)


We investigate stochastic differential equations with jumps and irregular coefficients, and obtain the existence and uniqueness ofgeneralized stochastic flows. Moreover, we also prove the existence and uniqueness of $L^p$-solutions or measure-valued solutionsfor second order integro-differential equation of Fokker-Planck type.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-25

Publication Date: May 20, 2013

DOI: 10.1214/EJP.v18-2820


  • Ambrosio, Luigi. Transport equation and Cauchy problem for $BV$ vector fields. Invent. Math. 158 (2004), no. 2, 227--260. MR2096794
  • Applebaum, David. Lévy processes and stochastic calculus. Cambridge Studies in Advanced Mathematics, 93. Cambridge University Press, Cambridge, 2004. xxiv+384 pp. ISBN: 0-521-83263-2 MR2072890
  • 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
  • Cipriano, Fernanda; Cruzeiro, Ana Bela. Flows associated with irregular $\Bbb R^ d$-vector fields. J. Differential Equations 219 (2005), no. 1, 183--201. MR2181034
  • Cruzeiro, Ana Bela. Équations différentielles ordinaires: non explosion et mesures quasi-invariantes. (French) [Ordinary differential equations: nonexplosion and quasi-invariant measures] J. Funct. Anal. 54 (1983), no. 2, 193--205. MR0724704
  • 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
  • Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8 MR0838085
  • Evans, Lawrence C.; Gariepy, Ronald F. Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. viii+268 pp. ISBN: 0-8493-7157-0 MR1158660
  • 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
  • Fournier, Nicolas. Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump. Electron. J. Probab. 13 (2008), no. 6, 135--156. MR2375602
  • Fujiwara, Tsukasa; Kunita, Hiroshi. Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group. J. Math. Kyoto Univ. 25 (1985), no. 1, 71--106. MR0777247
  • 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
  • Kulik A.: Stochastic calculus of variations for general Lévy processes and its applications to jump-type SDE's with non-degenerated drift.
  • Kunita, Hiroshi. Stochastic differential equations with jumps and stochastic flows of diffeomorphisms. Itô's stochastic calculus and probability theory, 197--211, Springer, Tokyo, 1996. MR1439526
  • Kunita, Hiroshi. Itô's stochastic calculus: its surprising power for applications. Stochastic Process. Appl. 120 (2010), no. 5, 622--652. MR2603057
  • Le Bris, C.; Lions, P.-L. Renormalized solutions of some transport equations with partially $W^ {1,1}$ velocities and applications. Ann. Mat. Pura Appl. (4) 183 (2004), no. 1, 97--130. MR2044334
  • 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
  • Lépingle, Dominique; Mémin, Jean. Sur l'intégrabilité uniforme des martingales exponentielles. (French) Z. Wahrsch. Verw. Gebiete 42 (1978), no. 3, 175--203. MR0489492
  • Lepeltier, J.-P.; Marchal, B. Problème des martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel. (French) Ann. Inst. H. Poincaré Sect. B (N.S.) 12 (1976), no. 1, 43--103. MR0413288
  • Li, Huaiqian; Luo, Dejun. Quasi-invariant flow generated by Stratonovich SDE with BV drift coefficient. Stoch. Anal. Appl. 30 (2012), no. 2, 258--284. MR2891455
  • Qiao, Huijie; Zhang, Xicheng. Homeomorphism flows for non-Lipschitz stochastic differential equations with jumps. Stochastic Process. Appl. 118 (2008), no. 12, 2254--2268. MR2474350
  • Protter, Philip E. Stochastic integration and differential equations. Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004. xiv+415 pp. ISBN: 3-540-00313-4 MR2020294
  • Protter P., Shimbo K.: No arbitrage and general semimartingales, http:// ~protter/ WebPapers/na-girsanov8.pdf.
  • Ren, Jie; Zhang, Xicheng. Limit theorems for stochastic differential equations with discontinuous coefficients. SIAM J. Math. Anal. 43 (2011), no. 1, 302--321. MR2765692
  • 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 MR1725357
  • 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
  • Situ, Rong. Theory of stochastic differential equations with jumps and applications. Mathematical and analytical techniques with applications to engineering. Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York, 2005. xx+434 pp. ISBN: 978-0387-25083-0; 0-387-25083-2 MR2160585
  • Stroock, Daniel W. Diffusion processes associated with Lévy generators. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), no. 3, 209--244. MR0433614
  • Stroock, Daniel W.; Varadhan, S. R. Srinivasa. Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233. Springer-Verlag, Berlin-New York, 1979. xii+338 pp. ISBN: 3-540-90353-4 MR0532498
  • Zhang, Xicheng. Stochastic flows of SDEs with irregular coefficients and stochastic transport equations. Bull. Sci. Math. 134 (2010), no. 4, 340--378. MR2651896
  • Zhang, Xicheng. Quasi-invariant stochastic flows of SDEs with non-smooth drifts on compact manifolds. Stochastic Process. Appl. 121 (2011), no. 6, 1373--1388. MR2794981
  • Zhang, Xicheng. Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients. Rev. Mat. Iberoam. 29 (2013), no. 1, 25--52. MR3010120

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