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