  For $\alpha >0$, the $\alpha$-Lipschitz minorant of a function $f :
\mathbb{R} \rightarrow \mathbb{R}$ is the greatest function $m : \mathbb{R}
\rightarrow \mathbb{R}$ such that $m \leq f$ and $\vert m(s) - m(t) \vert \leq
\alpha \vert s-t \vert$ for all $s,t \in \mathbb{R}$, should such a function
exist. If $X=(X_t)_{t \in \mathbb{R}}$ is a real-valued L\'evy process that is
not a pure linear drift with slope $\pm \alpha$, then the sample paths of $X$
have an $\alpha$-Lipschitz minorant almost surely if and only if
$\mathbb{E}[\vert X_1 \vert]< \infty$ and $\vert \mathbb{E}[X_1]\vert <
\alpha$. Denoting the minorant by $M$, we consider the contact set
$\mathcal{Z}:=\{ t \in \mathbb{R} : M_t = X_t \wedge X_{t-}\}$, which, since it
is regenerative and stationary, has the distribution of the closed range of
some subordinator "made stationary" in a suitable sense. We provide a
description of the excursions of the L\'evy process away from its contact set
similar to the one presented in It\^o excursion theory. We study the
distribution of the excursion on the special interval straddling zero. We also
give an explicit path decomposition of the other "generic" excursions in the
case of Brownian motion with drift $\beta$ with $\vert \beta \vert < \alpha$.
Finally, we investigate the progressive enlargement of the Brownian filtration
by the random time that is the first point of the contact set after zero.
