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The first-exit time process of an inverse Gaussian L\'evy process isconsidered. The one-dimensional distribution functions of the process areobtained. They are not infinitely divisible and the tail probabilities decayexponentially. These distribution functions can also be viewed as distributionfunctions of supremum of the Brownian motion with drift. The density functionis shown to solve a fractional PDE and the result is also generalized totempered stable subordinators. The subordination of this process to theBrownian motion is considered and the underlying PDE of the subordinatedprocess is obtained. The infinite divisibility of the first-exit time of a$\beta$-stable subordinator is also discussed.

First-Exit Times of an Inverse Gaussian Process