Lévy Flight vs Brownian Motion

Lévy flight uses step sizes drawn from a power-law stable distribution P(ℓ) ~ ℓ^−1−α (α∈(0,2)). For α<2, the variance diverges — individual long jumps dominate, giving superdiffusion: ⟨r²⟩ ~ t^γ with γ=2/α>1. At α=2, Lévy → Gaussian (Brownian); at α=1 → Cauchy. Green line shows a Brownian walker (α=2) for comparison. Lévy statistics describe animal foraging (Lévy foraging hypothesis), financial returns, earthquake epicenters, and light scattering in Lévy glasses.