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From Sturm–Liouville problems to fractional and anomalous diffusions

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  • D’Ovidio, Mirko

Abstract

Some fractional and anomalous diffusions are driven by equations involving fractional derivatives in both time and space. Such diffusions are processes with randomly varying times. In representing the solutions to those equations, the explicit laws of certain stable processes turn out to be fundamental. This paper directs one’s efforts towards the explicit representation of solutions to fractional and anomalous diffusions related to Sturm–Liouville problems of fractional order associated to fractional power function spaces. Furthermore, we study a new version of Bochner’s subordination rule and we establish some connections between subordination and space-fractional operators.

Suggested Citation

  • D’Ovidio, Mirko, 2012. "From Sturm–Liouville problems to fractional and anomalous diffusions," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3513-3544.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:10:p:3513-3544
    DOI: 10.1016/j.spa.2012.06.002
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    References listed on IDEAS

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    1. Beghin, Luisa & Orsingher, Enzo, 2009. "Iterated elastic Brownian motions and fractional diffusion equations," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1975-2003, June.
    2. 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.
    3. Metzler, Ralf & Klafter, Joseph, 2000. "Boundary value problems for fractional diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 278(1), pages 107-125.
    4. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2008. "Triangular array limits for continuous time random walks," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1606-1633, September.
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