The time T for a random walker to first reach an absorbing barrier follows P(T=t) ~ t−3/2 — a heavy-tailed Lévy-stable distribution with infinite mean when drift μ=0.
Parameters
Lévy-Smirnov distribution
P(T) ~ T−3/2 exp(−L²/2T)
μ=0: infinite mean first-passage time
μ>0: finite mean T* ≈ L/μ
μ<0: never reaches barrier (prob)
Red line = theoretical T−3/2 decay
on log-log scale.