Journal
JOURNAL OF STATISTICAL PHYSICS
Volume 141, Issue 6, Pages 1071-1092Publisher
SPRINGER
DOI: 10.1007/s10955-010-0086-6
Keywords
Continuous-time random-walk; Anomalous diffusion; Feynman-Kac; equation; Levy flights; Fractional calculus
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Funding
- Israel Science Foundation
- Israel Academy of Sciences and Humanities
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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 Schrodinger 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 L,vy 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 Schrodinger equations that recently appeared in the literature.
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