Potential Analysis

, Volume 23, Issue 1, pp 55–81

A Probabilistic Approach for Nonlinear Equations Involving the Fractional Laplacian and a Singular Operator

  • Benjamin Jourdain
  • Sylvie Méléard
  • Wojbor A. Woyczynski
Article

Abstract

We consider a class of nonlinear integro-differential equations involving a fractional power of the Laplacian and a nonlocal quadratic nonlinearity represented by a singular integral operator. Initially, we introduce cut-off versions of this equation, replacing the singular operator by its Lipschitz continuous regularizations. In both cases we show the local existence and global uniqueness in L1Lp. Then we associate with each regularized equation a stable-process-driven nonlinear diffusion; the law of this nonlinear diffusion has a density which is a global solution in L1 of the cut-off equation. In the next step we remove the cut-off and show that the above densities converge in a certain space to a solution of the singular equation. In the general case, the result is local, but under a more stringent balance condition relating the dimension, the power of the fractional Laplacian and the degree of the singularity, it is global and gives global existence for the original singular equation. Finally, we associate with the singular equation a nonlinear singular diffusion and prove propagation of chaos to the law of this diffusion for the related cut-off interacting particle systems. Depending on the nature of the singularity in the drift term, we obtain either a strong pathwise result or a weak convergence result.

Keywords

propagation of chaos nonlinear stochastic differential equations driven by Lévy processes partial differential equation with fractional Laplacian nonlinear singular operator 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aldous, D.: ‘Stopping times and tightness’, Ann. Probab. 6 (1978), 335–340. Google Scholar
  2. 2.
    Bichteler, K., Gravereaux, J.B. and Jacod, J.: Malliavin Calculus for Processes with Jumps, Stochastics Monographs 2, Gordon and Breach Science Publishers, 1987. Google Scholar
  3. 3.
    Biler, P. and Nadzieja, T.: ‘Existence and nonexistence of solutions for a model of gravitational interaction of particles’, Colloquium Math. 66 (1994), 319–334. Google Scholar
  4. 4.
    Biler, P. and Nadzieja, T.: ‘A singular problem in electrolytes theory’, Math. Methods Appl. Sci. 20 (1997), 767–782. Google Scholar
  5. 5.
    Biler, P. and Woyczynski, W.A.: ‘Global and exploding solutions for nonlocal quadratic evolution problems’, SIAM J. Appl. Math. 59 (1998), 845–869. Google Scholar
  6. 6.
    Biler, P., Funaki, T. and Woyczynski, W.A.: ‘Interacting particle approximations for nonlocal quadratic evolution problems’, Probab. Math. Statist. 19(2) (1999), 267–286. Google Scholar
  7. 7.
    Biler, P., Karch, G. and Woyczynski, W.A.: ‘Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws’, Ann. Inst. Henri Poincaré Nonlinear Anal. 18 (2001), 613–637. Google Scholar
  8. 8.
    Cannone, M.: Ondelettes, Paraproduits et Navier–Stokes, Diderot Editeur, Paris, 1995. Google Scholar
  9. 9.
    Dobrushin, R.L.: ‘Prescribing a system of random variables by conditional expectations’, Theory Probab. Appl. 15(3) (1970), 450. Google Scholar
  10. 10.
    Ethier, S.N. and Kurtz, T.G.: Markov Processes, Characterization and Convergence, Wiley, 1986. Google Scholar
  11. 11.
    Fontbona, J.: ‘Nonlinear martinagle problems involving singular integrals’, J. Funct. Anal. 200(1) (2003), 198–236. Google Scholar
  12. 12.
    Jourdain, B. and Méléard, S.: ‘Propagation of chaos and fluctuations for a moderate model with smooth initial data’, Ann. Inst. Henri Poincaré Probab. 34(6) (1998), 727–766. Google Scholar
  13. 13.
    Jourdain, B.: ‘Diffusion processes associated with nonlinear evolution equations for signed measures’, Methodology and Computing in Applied Probability 2(1) (2000), 69–91. Google Scholar
  14. 14.
    Mann, J.A. and Woyczynski, W.A.: ‘Growing fractal interfaces in the presence of self-similar hopping surface diffusion’, Physica A 291 (2001), 159–183. Google Scholar
  15. 15.
    Liggett, T.: Interacting Particle Systems, Springer, 1985. Google Scholar
  16. 16.
    Méléard, S.: ‘Asymptotic behaviour of some interacting particle systems: McKean–Vlasov and Boltzmann models’, in CIME 1995, Probabilistic Models for Nonlinear Partial Differential Equations, Lecture Notes in Math. 1627, Springer, 1996, pp. 42–95. Google Scholar
  17. 17.
    Méléard, S.: ‘A trajectorial proof of the vortex method for the two-dimensional Navier–Stokes equation’, Ann. Appl. Probab. 10(4) (2000), 1197–1211. Google Scholar
  18. 18.
    Méléard, S.: ‘Monte-Carlo approximations for 2d Navier–Stokes equations with measure initial data’, Probab. Theory Related Fields 121 (2001), 367–388. Google Scholar
  19. 19.
    Metzler, R. and Klafter, J.: ‘The random walk’s guide to anomalous diffusion: A fractional dynamics approach’, Phys. Rep. 339 (2000), 1–77. CrossRefMATHGoogle Scholar
  20. 20.
    Saichev, A.I. and Woyczynski, W.A.: Distributions in the Physical and Engineering Sciences. Volume 2: Linear, Nonlinear, Random and Fractal Dynamics in Continuous Media, Birkhäuser, Boston, 2004. Google Scholar
  21. 21.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Math. 68, Cambridge University Press, 1999. Google Scholar
  22. 22.
    Stroock, D.W. and Varadhan, S.R.S.: Multidimensional Diffusion Processes, Springer-Verlag, 1979. Google Scholar
  23. 23.
    Sznitman, A.S.: Topics in Propagation of Chaos, Ecole d’été de probabilités de Saint-Flour XIX – 1989, Lecture Notes in Math. 1464, Springer-Verlag, 1991. Google Scholar
  24. 24.
    Zheng, W.: ‘Conditional propagation of chaos and a class of quasilinear PDE’s’, Ann. Probab. 23 (1995), 1389–1413. Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Benjamin Jourdain
    • 1
  • Sylvie Méléard
    • 2
  • Wojbor A. Woyczynski
    • 3
  1. 1.ENPC-CERMICSMarne la Vallée Cedex 2France
  2. 2.Université Paris 10, MODALXNanterreFrance
  3. 3.Department of Statistics and Center for Stochastic and Chaotic Processes in Science and TechnologyCase Western Reserve UniversityClevelandU.S.A.

Personalised recommendations