IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v115y2005i6p1017-1040.html
   My bibliography  Save this article

The distribution of the local time for "pseudoprocesses" and its connection with fractional diffusion equations

Author

Listed:
  • Beghin, L.
  • Orsingher, E.

Abstract

We prove that the pseudoprocesses governed by heat-type equations of order n[greater-or-equal, slanted]2 have a local time in zero (denoted by ) whose distribution coincides with the folded fundamental solution of a fractional diffusion equation of order 2(n-1)/n, n[greater-or-equal, slanted]2. The distribution of is also expressed in terms of stable laws of order n/(n-1) and their form is analyzed. Furthermore, it is proved that the distribution of is connected with a wave equation as n-->[infinity]. The distribution of the local time in zero for the pseudoprocess related to the Myiamoto's equation is also derived and examined together with the corresponding telegraph-type fractional equation.

Suggested Citation

  • Beghin, L. & Orsingher, E., 2005. "The distribution of the local time for "pseudoprocesses" and its connection with fractional diffusion equations," Stochastic Processes and their Applications, Elsevier, vol. 115(6), pages 1017-1040, June.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:6:p:1017-1040
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(05)00015-3
    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. Hochberg, Kenneth J. & Orsingher, Enzo, 1994. "The arc-sine law and its analogs for processes governed by signed and complex measures," Stochastic Processes and their Applications, Elsevier, vol. 52(2), pages 273-292, August.
    2. Beghin, L. & Orsingher, E. & Ragozina, T., 2001. "Joint distributions of the maximum and the process for higher-order diffusions," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 71-93, July.
    3. Beghin, Luisa & Hochberg, Kenneth J. & Orsingher, Enzo, 2000. "Conditional maximal distributions of processes related to higher-order heat-type equations," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 209-223, February.
    Full references (including those not matched with items on IDEAS)

    Citations

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


    Cited by:

    1. Aimé Lachal, 2012. "A Survey on the Pseudo-process Driven by the High-order Heat-type Equation $\boldsymbol{\partial/\partial t=\pm\partial^N\!/\partial x^N}$ Concerning the Hitting and Sojourn Times," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 549-566, September.
    2. Ibragimov, I.A. & Smorodina, N.V. & Faddeev, M.M., 2015. "Limit theorems for symmetric random walks and probabilistic approximation of the Cauchy problem solution for Schrödinger type evolution equations," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4455-4472.
    3. Lachal, Aimé, 2014. "First exit time from a bounded interval for pseudo-processes driven by the equation ∂/∂t=(−1)N−1∂2N/∂x2N," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1084-1111.
    4. Mura, A. & Taqqu, M.S. & Mainardi, F., 2008. "Non-Markovian diffusion equations and processes: Analysis and simulations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(21), pages 5033-5064.

    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. Lachal, Aimé, 2014. "First exit time from a bounded interval for pseudo-processes driven by the equation ∂/∂t=(−1)N−1∂2N/∂x2N," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1084-1111.
    2. Aimé Lachal, 2012. "A Survey on the Pseudo-process Driven by the High-order Heat-type Equation $\boldsymbol{\partial/\partial t=\pm\partial^N\!/\partial x^N}$ Concerning the Hitting and Sojourn Times," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 549-566, September.
    3. Beghin, L. & Orsingher, E. & Ragozina, T., 2001. "Joint distributions of the maximum and the process for higher-order diffusions," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 71-93, July.
    4. Bonaccorsi, Stefano & Mazzucchi, Sonia, 2015. "High order heat-type equations and random walks on the complex plane," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 797-818.
    5. Lachal, Aimé, 2008. "First hitting time and place for pseudo-processes driven by the equation subject to a linear drift," Stochastic Processes and their Applications, Elsevier, vol. 118(1), pages 1-27, January.
    6. D’Ovidio, Mirko, 2011. "On the fractional counterpart of the higher-order equations," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1929-1939.
    7. Y. Nikitin & E. Orsingher, 2000. "On Sojourn Distributions of Processes Related to Some Higher-Order Heat-Type Equations," Journal of Theoretical Probability, Springer, vol. 13(4), pages 997-1012, October.
    8. Beghin, Luisa & Hochberg, Kenneth J. & Orsingher, Enzo, 2000. "Conditional maximal distributions of processes related to higher-order heat-type equations," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 209-223, February.
    9. van Noortwijk, J.M. & van der Weide, J.A.M. & Kallen, M.J. & Pandey, M.D., 2007. "Gamma processes and peaks-over-threshold distributions for time-dependent reliability," Reliability Engineering and System Safety, Elsevier, vol. 92(12), pages 1651-1658.
    10. D'Ovidio, Mirko & Orsingher, Enzo, 2011. "Bessel processes and hyperbolic Brownian motions stopped at different random times," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 441-465, March.
    11. Bonaccorsi, Stefano & Calcaterra, Craig & Mazzucchi, Sonia, 2017. "An Itô calculus for a class of limit processes arising from random walks on the complex plane," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2816-2840.

    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:115:y:2005:i:6:p:1017-1040. 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.