IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v118y2008i2p284-318.html
   My bibliography  Save this article

A wavelet particle approximation for McKean-Vlasov and 2D-Navier-Stokes statistical solutions

Author

Listed:
  • Tran, Viet Chi

Abstract

Letting the initial condition of a PDE be random is interesting when considering complex phenomena. For 2D-Navier-Stokes equations, it is for instance an attempt to take into account the turbulence arising with high velocities and low viscosities. The solutions of these PDEs are random and their laws are called statistical solutions. We start by studying McKean-Vlasov equations with initial conditions parameterized by a real random variable [theta], and link their weak measure solutions to the laws of nonlinear SDEs, for which the drift coefficients are expressed as conditional expectations in the diffusions' laws given [theta]. We propose an original stochastic particle method to compute the first-order moments of the statistical solutions, obtained by approximating the conditional expectations by wavelet regression estimators. We establish a convergence rate that improves the ones obtained for existing methods with Nadaraya-Watson kernel estimators. We then carry over these results to 2D-Navier-Stokes equations and compute some physical quantities of interest, like the mean velocity vector field. Numerical simulations illustrate the method and allow us to test its robustness.

Suggested Citation

  • Tran, Viet Chi, 2008. "A wavelet particle approximation for McKean-Vlasov and 2D-Navier-Stokes statistical solutions," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 284-318, February.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:2:p:284-318
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(07)00067-1
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. B. Jourdain, 2000. "Diffusion Processes Associated with Nonlinear Evolution Equations for Signed Measures," Methodology and Computing in Applied Probability, Springer, vol. 2(1), pages 69-91, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Shkolnikov, Mykhaylo, 2012. "Large systems of diffusions interacting through their ranks," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1730-1747.
    2. Shkolnikov, Mykhaylo, 2013. "Large volatility-stabilized markets," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 212-228.
    3. Rémillard, Bruno & Vaillancourt, Jean, 2014. "On signed measure valued solutions of stochastic evolution equations," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 101-122.
    4. BOSSY Mireille & JOURDAIN Benjamin, 2001. "A Stochastic Particle Method For The Solution Of A 1d Viscous Scalar Conservation Law In A Bounded Interval," Monte Carlo Methods and Applications, De Gruyter, vol. 7(1-2), pages 45-54, December.
    5. Kolli, Praveen & Sarantsev, Andrey, 2019. "Large rank-based models with common noise," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 29-35.
    6. Andrey Sarantsev, 2019. "Comparison Techniques for Competing Brownian Particles," Journal of Theoretical Probability, Springer, vol. 32(2), pages 545-585, June.
    7. Jourdain, B., 2000. "Probabilistic approximation for a porous medium equation," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 81-99, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:118:y:2008:i:2:p:284-318. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.