📂Probability Theory

Lévy Flight

Definition ^{1 2}

If the stepwise increment $X_{t} - X_{t-1}$ of a random walk $X_{t}$ is distributed according to a heavy-tailed stable distribution, the process is called a Lévy flight.

Description

A Lévy flight is a stochastic process, typically in Euclidean space, in which the motion of a point exhibits intermittent large jumps. While Brownian motion is merely diffusive, a Lévy flight exhibits superdiffusion. These jumps are intrinsic to the stochastic process itself, unlike concepts such as outliers, and there are applications in explaining or simulating large-scale migrations in ecosystems, e.g. migratory birds or locust swarms.

Simulation

Seeing is believing — it is easiest to understand by watching. The following is a direct simulation of a Lévy flight in two dimensions with a total of 1000 timesteps.

As shown, the trajectory typically moves in short segments but occasionally leaves extremely long line segments and jumps far away. By contrast, the following is the familiar Brownian motion. To get a sense of how far they have moved by the end of the simulation, check the units of the coordinates.

Self-similarity ^{2 3}

A Lévy flight is a self-similar stochastic process; its trajectory has fractal dimension $\alpha$ when the probability density function (PDF) of the increments $x$ is of the form $f$ for sufficiently large arguments. $$ f(x) \sim {\frac{ 1 }{ x^{1 + \alpha} }} $$

Code

The following is simulation code written in Julia.

using Distributions, LinearAlgebra, Plots
default(legend = :none, color = :black, aspect_ratio = 1)
MvCauchy = MvTDist(1, zeros(2), Matrix(I(2)))

tend = 1000
XY1 = cumsum(rand(MvCauchy, tend), dims = 2)
anime1 = @animate for t in 1:tend
    plot(XY1[1, 1:t], XY1[2, 1:t], size = [400, 400])
end
gif(anime1, "levy.mp4")

XY2 = cumsum(randn(2, tend), dims = 2)
anime2 = @animate for t in 1:tend
    plot(XY2[1, 1:t], XY2[2, 1:t], size = [400, 400])
end
gif(anime2, "brownian.mp4")

One caution in the simulation: the bivariate Cauchy distribution should not be sampled by drawing Cauchy samples coordinate-wise (unlike the multivariate normal distribution). Instead, sample from the multivariate $t$-distribution with degrees of freedom equal to 1.