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

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  • 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
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    References listed on IDEAS

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    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.
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    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.

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