Skip to main content
SpringerLink
Log in

Non-local Conservation Law from Stochastic Particle Systems

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

We consider an interacting particle system in \(\mathbb {R}^d\) modelled as a system of N stochastic differential equations. The limiting behaviour as the size N grows to infinity is achieved as a law of large numbers for the empirical density process associated with the interacting particle system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The proof is elementary, using the fact that if a set is compact in \(\mathbb {L}^{2}\big ( [0,T]\; ;\; \mathbb {W}_\mathrm{loc}^{\eta ,2} ( \mathcal {B} ( 0,n ) ) \big ) \) for every n then it is compact in \(\mathbb {L}^{2}\big ( [0,T]\; ;\; \mathbb {W}_\mathrm{loc}^{\varepsilon ,2}( \mathbb {R}^{d}) \big ) \) with this topology.

References

  1. Alibaud, N.: Entropy formulation for fractal conservation laws. J. Evol. Equ. 7, 145–175 (2007)

    Article  MathSciNet  Google Scholar 

  2. Alfaro, M., Droniou, J.: General fractal conservation laws arising from a model of detonations in gases. Appl. Math. Res. eXpress 2012(2), 127–151 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Andreu, F., Mazon, J.M., Rossi, J.D., Toledo, J.: Nonlocal Diffusion Problems. AMS Mathematical Surveys and Monographs, vol. 165. AMS, Providence, RI (2010)

  4. Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  5. Biler, P., Funaki, T., Woyczynski, W.A.: Interacting particle approximations for nonlocal quadratic evolution problems. Prob. Math. Stat. 2, 267–286 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Capasso, V., Bakstein, D.: An Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology and Medicine (Modeling and Simulation in Science, Engineering and Technology). Springer, Birkhäuser, New York, Basel (2015)

    Book  Google Scholar 

  7. Clavin, P.: Instabilities and nonlinear patterns of overdriven detonations in gases. In: Berestycki, H., Pomeau, Y. (eds.) Nonlinear PDE’s in Condensed Matter and Reactive Flows, pp. 49–97. Kluwer (2002)

  8. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  9. Droniou, J., Imbert, C.: Fractal first-order partial differential equations. Arch. Ration. Mech. Anal. 182, 299–331 (2006)

    Article  MathSciNet  Google Scholar 

  10. Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier–Stokes equations. Prob. Theory Relat. Fields 102, 367–391 (1995)

    Article  MathSciNet  Google Scholar 

  11. Flandoli, F., Leimbach, M., Olivera, C.: Uniform approximation of FKPP equation by stochastic particle systems. ArXiv:1604.03055 (2016)

  12. Gyongy, I., Krylov, N.: Existence of strong solutions for Itô stochastic equations, via approximations. Prob. Theory Relat. Fields 105, 143–158 (1996)

    Article  Google Scholar 

  13. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North Holland Pub. Co., Amsterdam (1981)

    MATH  Google Scholar 

  14. Jourdain, B., Méléard, S., Woyczynski, W.A.: A probabilistic approach for non-linear equations involving fractional Laplacian and singular operator. Potential Anal. 23(1), 55–81 (2005)

    Article  MathSciNet  Google Scholar 

  15. Jourdain, B., Méléard, S., Woyczynski, W.A.: Probabilistic approximation and inviscid limits for 1-D fractional conservation laws. Bernoulli 11(4), 689–714 (2005)

    Article  MathSciNet  Google Scholar 

  16. Jourdain, B., Méléard, S.: Propagation of chaos and fluctuations for a moderate model with smooth initial data. Ann. IHP (B) Probab. Stat. 34(6), 727–766 (1998)

    MathSciNet  MATH  Google Scholar 

  17. Karch, G.: Non-linear evolution equations with anomalous diffusion. In: Qualitative Properties of Solutions to Partial Differential Equations, Jindrich Necas Center for Mathematical Modeling Lecture Notes, vol. 5, pp. 2–68. Matfyzpress, Prague (2009)

  18. Kunita, H.: Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms. In: Rao, M.M. (ed.) Real and Stochastic Analysis, New Perspective, pp. 305–374. Birkhäuser (2004)

  19. Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  20. Méléard, S., Roelly-Coppoletta, S.: A propagation of chaos result for a system of particles with moderate interaction. Stoch. Process. Appl. 26, 317–332 (1987)

    Article  MathSciNet  Google Scholar 

  21. Oelschläger, K.: A law of large numbers for moderately interacting diffusion processes. Zeitschrift fur Wahrsch. Verwandte Gebiete 69, 279–322 (1985)

    Article  MathSciNet  Google Scholar 

  22. Priola, E.: Pathwise uniqueness for singular SDEs drive n by stable processes. Osaka J. Math. 49(2), 421–447 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Sato, K.: Lévy Processes and Infinitely Divisible Distributions (Cambridge Studies in Advanced Mathematics), 2nd edn. Cambridge University Press (2013)

  24. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  25. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, New York (1978)

  26. Varadhan, S.R.S.: Scaling limits for interacting diffusions. Commun. Math. Phys. 135, 313–353 (1991)

    Article  MathSciNet  Google Scholar 

  27. Vazquez, J.L.: Recent progress in the theory of non-linear diffusion with fractional Laplacian operators. Discrete Contin. Dyn. Syst. Ser. S 7, 857–885 (2014)

    Article  MathSciNet  Google Scholar 

  28. Woyczynski, W.: Lévy Processes in the Physical Sciences, pp. 241–266. Birkhäuser, Boston (2001)

    Book  Google Scholar 

  29. Zhang, X.: \(L^{p}\)-maximal regularity of nonlocal parabolic equation and applications. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 30(4), 573–614 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

C.O. is partially supported by CNPq through the Grant 460713/2014-0 and FAPESP by the Grants 2015/04723-2 and 2015/07278-0. This work benefited from the support of the Project EDNHS ANR-14-CE25-0011 of the French National Research Agency (ANR). The work of M.S. was also supported by the Labex CEMPI (ANR-11-LABX-0007-01).

Author information

Authors and Affiliations

  1. MEPHYSTO Team, INRIA Lille - Nord Europe, Villeneuve-d’Ascq, France

    Marielle Simon

  2. Laboratoire Paul Painlevé, CNRS UMR, 8524, Villeneuve-d’Ascq, France

    Marielle Simon

  3. Departamento de Matemática, Universidade Estadual de Campinas, Campinas, Brazil

    Christian Olivera

Authors
  1. Marielle Simon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Olivera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Olivera.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Simon, M., Olivera, C. Non-local Conservation Law from Stochastic Particle Systems. J Dyn Diff Equat 30, 1661–1682 (2018). https://doi.org/10.1007/s10884-017-9620-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-017-9620-4

Keywords

Mathematics Subject Classification

Search

Navigation