Lévy flight

A Lévy flight is a random walk in which the step-lengths have a stable distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions. Later researchers have extended the use of the term "Lévy flight" to also include cases where the random walk takes place on a discrete grid rather than on a continuous space. The term "Lévy flight" was coined after Paul Lévy by Benoît Mandelbrot, who used this for one specific definition of the distribution of step sizes. He used the term Cauchy flight for the case where the distribution of step sizes is a Cauchy distribution, and Rayleigh flight for when the distribution is a normal distribution (which is not an example of a heavy-tailed probability distribution). The particular case for which Mandelbrot used the term "Lévy flight" is defined by the survival function of the distribution of step-sizes, U, being

Pr ( U > u ) = { 1 : u < 1 , u − D : u ≥ 1. {\displaystyle \Pr(U>u)={\begin{cases}1&:\ u<1,\\u^{-D}&:\ u\geq 1.\end{cases}}}

Here D is a parameter related to the fractal dimension and the distribution is a particular case of the Pareto distribution.

Properties Lévy flights are, by construction, Markov processes. For general distributions of the step-size, satisfying the power-like condition, the distance from the origin of the random walk tends, after a large number of steps, to a stable distribution due to the generalized central limit theorem, enabling many processes to be modeled using Lévy flights. The probability densities for particles undergoing a Levy flight can be modeled using a generalized version of the Fokker–Planck equation, which is usually used to model Brownian motion. The equation requires the use of fractional derivatives. For jump lengths which have a symmetric probability distribution, the equation takes a simple form in terms of the Riesz fractional derivative. In one dimension, the equation reads as

∂ φ ( x , t ) ∂ t = − ∂ ∂ x f ( x , t ) φ ( x , t ) + γ ∂ α φ ( x , t ) ∂ | x | α {\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=-{\frac {\partial }{\partial x}}f(x,t)\varphi (x,t)+\gamma {\frac {\partial ^{\alpha }\varphi (x,t)}{\partial |x|^{\alpha }}}}

where γ is a constant akin to the diffusion constant, α is the stability parameter and f(x,t) is the potential. The Riesz derivative can be understood in terms of its Fourier transform.

F k [ ∂ α φ ( x , t ) ∂ | x | α ] = − | k | α F k [ φ ( x , t ) ] {\displaystyle F_{k}\left[{\frac {\partial ^{\alpha }\varphi (x,t)}{\partial |x|^{\alpha }}}\right]=-|k|^{\alpha }F_{k}[\varphi (x,t)]}

This can be easily extended to multiple dimensions. Another important property of the Lévy flight is that of diverging variances in all cases except that of α = 2, i.e. Brownian motion. In general, the θ fractional moment of the distribution diverges if α ≤ θ. Also,

⟨ | x | θ ⟩ ∝ t θ / α if θ < α . {\displaystyle \left\langle |x|^{\theta }\right\rangle \propto t^{\theta /\alpha }\quad {\text{if }}\theta <\alpha .}

The exponential scaling of the step lengths gives Lévy flights a scale invariant property, and they are used to model data that exhibits clustering.

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