the first-passage time law
How long does Brownian motion take to reach a level a for the first time? The answer is the first-passage (hitting) time tau_a = inf{ t : B_t = a }, and its law is one of the most important explicit distributions in the theory — a one-sided stable law with index 1/2, with tails so heavy that tau_a has INFINITE mean despite being almost surely finite.
By the reflection principle, P(tau_a <= t) = P(M_t >= a) = 2 P(B_t >= a) = 2(1 - Phi(a / sqrt(t))). Differentiating in t gives the density f_(tau_a)(t) = (a / sqrt(2 pi t^3)) exp(-a^2 / (2t)) for t > 0; this is the Levy (stable-1/2) distribution, also the inverse-Gaussian limit. Two facts make it remarkable. First, tau_a < infinity almost surely (Brownian motion is recurrent on the line, so it reaches every level), yet E[tau_a] = +infinity because the density decays only like t^(-3/2) at large t. Second, the family {tau_a : a >= 0} has stationary independent increments — by the strong Markov property the time to go from a to a + b after hitting a is an independent copy of tau_b — so a -> tau_a is itself a Levy process, in fact the stable subordinator of index 1/2. Its Laplace transform is E[exp(-lambda tau_a)] = exp(-a sqrt(2 lambda)), the clean signature of the 1/2-stable law.
Why it matters: first-passage laws govern ruin and default times, the timing of barrier hits in finance, and the scaling tau_a =d a^2 tau_1 that re-expresses Brownian self-similarity (distance scales like sqrt(time)). The infinite-mean tail is the honest content: you WILL hit level a, but the expected waiting time is infinite, so sample-path intuitions based on averages are misleading. For a > 0 the variable B_(tau_a) = a is deterministic, but tau_a is heavy-tailed; for two-sided exit from an interval the law is different (a fast-decaying series), so do not confuse one-sided hitting with two-sided exit.
Because tau_a =d a^2 tau_1, doubling the target level a quadruples the typical hitting time — Brownian distance grows like the square root of time. Yet E[tau_a] = infinity, so a fund that 'always eventually' recovers to a target can have infinite expected recovery time.
The hitting time is a stable-1/2 law: almost surely finite, heavy-tailed with density ~ t^(-3/2), and of infinite mean.
tau_a is finite almost surely but has E[tau_a] = infinity; and a -> tau_a is the 1/2-stable subordinator. One-sided hitting and two-sided exit from an interval obey DIFFERENT laws.