Journal of Statistical Physics

, Volume 141, Issue 6, pp 1071–1092 | Cite as

On Distributions of Functionals of Anomalous Diffusion Paths

  • Shai Carmi
  • Lior Turgeman
  • Eli Barkai


Functionals of Brownian motion have diverse applications in physics, mathematics, and other fields. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, which is a Schrödinger equation in imaginary time. In recent years there is a growing interest in particular functionals of non-Brownian motion, or anomalous diffusion, but no equation existed for their PDF. Here, we derive a fractional generalization of the Feynman-Kac equation for functionals of anomalous paths based on sub-diffusive continuous-time random walk. We also derive a backward equation and a generalization to Lévy flights. Solutions are presented for a wide number of applications including the occupation time in half space and in an interval, the first passage time, the maximal displacement, and the hitting probability. We briefly discuss other fractional Schrödinger equations that recently appeared in the literature.


Continuous-time random-walk Anomalous diffusion Feynman-Kac equation Levy flights Fractional calculus 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Physics and Advanced Materials and Nanotechnology InstituteBar-Ilan UniversityRamat GanIsrael

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