Brownian motion waiting time equations

The fact that the temporal occurrence of the motion events performed by the random walker is so broadly distributed that no characteristic waiting time exists has been exploited by a number of investigators [7,19,31] in order to generalize the various diffusion equations of Brownian dynamics to explain anomalous relaxation phenomena. The resulting diffusion equations are called fractional diffusion equations because in general they will involve fractional derivatives of the probability density with respect to the time. For example, in fractional noninertial diffusion in a potential, the diffusion Eq. (5) becomes [7,31]... 

We conclude that as long as the mean waiting time and the variance of the jumps are finite, parabolic scaling leads to the Brownian motion in the limit e 0. The macroscopic equation for the density of particles is a scale-invariant diffusion equation. Infinite variance of jumps in the domain of attraction of a stable law leads to Ldvy processes, Levy flights. In the limit e 0, the particle position X (t) becomes self-similar with exponent 1/a. Recall that the random process X(t) is self-similar, if there exists a scaling exponent H such that X t) and e X(t/e) have the same distributions for any scaling parameter e. In this case we write... 

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