IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v324y2003i3p530-550.html
   My bibliography  Save this article

Diffusion and memory effects for stochastic processes and fractional Langevin equations

Author

Listed:
  • Bazzani, Armando
  • Bassi, Gabriele
  • Turchetti, Giorgio

Abstract

We consider the diffusion processes defined by stochastic differential equations when the noise is correlated. A functional method based on the Dyson expansion for the evolution operator, associated to the stochastic continuity equation, is proposed to obtain the Fokker–Planck equation, after averaging over the stochastic process. In the white noise limit the standard result, corresponding to the Stratonovich interpretation of the non-linear Langevin equation, is recovered. When the noise is correlated the averaged operator series cannot be summed, unless a family of time-dependent operators commutes. In the case of a linear equation, the constraints are easily worked out. The process defined by a linear Langevin equation with additive noise is Gaussian and the probability density function of its fluctuating component satisfies a Fokker–Planck equation with a time-dependent diffusion coefficient. The same result holds for a linear Langevin equation with a fractional time derivative (defined according to Caputo, Elasticità e Dissipazione, Zanichelli, Bologna, 1969). In the generic linear or non-linear case approximate equations for small noise amplitude are obtained. For small correlation time the evolution equations further simplify in agreement with some previous alternative derivations. The results are illustrated by the linear oscillator with coloured noise and the fractional Wiener process, where the numerical simulation for the probability density and its moments is compared with the analytical solution.

Suggested Citation

  • Bazzani, Armando & Bassi, Gabriele & Turchetti, Giorgio, 2003. "Diffusion and memory effects for stochastic processes and fractional Langevin equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(3), pages 530-550.
    • Handle: RePEc:eee:phsmap:v:324:y:2003:i:3:p:530-550
    DOI: 10.1016/S0378-4371(03)00073-6
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437103000736
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/S0378-4371(03)00073-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

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

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Buyukkilic, F. & Ok Bayrakdar, Z. & Demirhan, D., 2015. "Investigation of cumulative growth process via Fibonacci method and fractional calculus," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 237-244.
    2. Bassi, Gabriele & Bazzani, Armando & Mais, Helmut & Turchetti, Giorgio, 2005. "Stochastic continuity equation and related processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 347(C), pages 17-37.
    3. Tawfik, Ashraf M. & Elkamash, I.S., 2022. "On the correlation between Kappa and Lévy stable distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).
    4. Drozdov, A.D., 2007. "Fractional oscillator driven by a Gaussian noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 376(C), pages 237-245.
    5. Piryatinska, A. & Saichev, A.I. & Woyczynski, W.A., 2005. "Models of anomalous diffusion: the subdiffusive case," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 349(3), pages 375-420.
    6. Enrica Pirozzi, 2024. "Mittag–Leffler Fractional Stochastic Integrals and Processes with Applications," Mathematics, MDPI, vol. 12(19), pages 1-20, October.
    7. Abundo, Mario & Pirozzi, Enrica, 2018. "Integrated stationary Ornstein–Uhlenbeck process, and double integral processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 494(C), pages 265-275.
    8. Wang, JinRong & Li, Xuezhu, 2015. "Ulam–Hyers stability of fractional Langevin equations," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 72-83.

    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:phsmap:v:324:y:2003:i:3:p:530-550. 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.

    We have no bibliographic references for this item. You can help adding them by using 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.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    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.