  We study boundary traces of shift-invariant diffusions: two-dimensional
diffusions in the upper half-plane $\mathbb{R} \times [0, \infty)$ (or in
$\mathbb{R} \times [0, R)$) invariant under horizontal translations. We prove
that the corresponding trace processes are L\'evy processes with completely
monotone jumps, and, conversely, every L\'evy process with completely monotone
jumps is a boundary trace of some shift-invariant diffusion. Up to some natural
transformations of space and time, this correspondence is bijective. We also
reformulate this result in the language of additive functionals of the Brownian
motion in $[0, \infty)$ (or in $[0, R)$), and Brownian excursions. Our main
tool is the recent extension of Krein's spectral theory of strings, due to
Eckhardt and Kostenko.

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