the inverse-Gaussian process
The inverse-Gaussian process is the subordinator whose increments are inverse-Gaussian distributed — and the name is literal: it is the process of first-passage times of a Brownian motion with drift to successive levels. It is a natural increasing random clock and, used to subordinate Brownian motion, it produces the normal-inverse-Gaussian (NIG) model, a flexible finance model competing with variance-gamma.
Concretely, if W_t = mu t + B_t is Brownian motion with positive drift mu, then T_a = inf{t : W_t = a}, the first time it reaches level a, defines an increasing process in a; the increment T_{a+da} - T_a is inverse-Gaussian. As a Lévy process T_a has Laplace exponent Phi(u) = delta(sqrt(2u + gamma^2) - gamma) (with parameters delta, gamma > 0), Lévy measure nu(dx) = (delta / sqrt(2 pi)) x^(-3/2) e^(-gamma^2 x / 2) dx on x > 0, no drift, no Gaussian part. The x^(-3/2) singularity at zero makes nu infinite (infinitely many small jumps) but integrable against x, so the process is finite variation, like the gamma process. The inverse-Gaussian law is itself the canonical first-passage distribution, infinitely divisible, with explicit density and moment generating function.
Subordinating Brownian motion by an IG process gives the normal-inverse-Gaussian (NIG) Lévy process of Barndorff-Nielsen, which has semi-heavy (exponentially-tilted) tails and is heavily used to model financial returns and turbulence. The honest point: 'inverse-Gaussian' does not mean the reciprocal of a Gaussian — the name comes from the inverse relationship between the cumulant generating functions of the first-passage time and the Gaussian, a historical naming that regularly confuses newcomers.
Let W_t = t + B_t hit level a for the first time at T_a. Then T_a is inverse-Gaussian with mean a (the time to drift distance a) and the process a -> T_a is the IG subordinator. Subordinating an independent Brownian motion B'_{T_a} yields the NIG process, whose tails decay exponentially-times-polynomially — lighter than stable, heavier than Gaussian.
First-passage times of drifting Brownian motion form the IG subordinator.
Inverse-Gaussian does NOT mean 1/Gaussian; the name reflects an inverse relationship between cumulant generating functions. The IG process has infinite Lévy measure but finite variation, and like all subordinators has no Gaussian part.