Levy flight processes Brownian motion

The main objective of this chapter is to establish the relation between the macroscopic equations like (3.1) and (3.5), the mesoscopic equations (3.2) and (3.3), etc., and the underlying microscopic movement of particles. We will show how to derive mesoscopic reaction-transport equations like (3.2) and (3.3) from microscopic random walk models. In particular, we will discuss the scaling procedures that lead to macroscopic reaction-transport equations. As an example, let us mention that the macroscopic reaction-diffusion equation (3.1) occurs as a result of the convergence of the random microscopic movement of particles to Brownian motion, while the macroscopic fractional equation (3.5) is closely related to the convergence of random walks with heavy-tailed jump PDFs to a-stable random processes or Levy flights. 

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... 

---