Lipschitz minorants of Brownian Motion and Levy processes

For $\alpha > 0$, the $\alpha$-Lipschitz minorant of a function $f: \mathbb{R} \to \mathbb{R}$ is the greatest function $m : \mathbb{R} \to \mathbb{R}$ such that $m \leq f$ and $|m(s)-m(t)| \le \alpha |s-t|$ 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 pure linear drift with slope $\pm \alpha$, then the sample paths of $X$ have an $\alpha$-Lipschitz minorant almost surely if and only if $| \mathbb{E}[X_1] | < \alpha$. Denoting the minorant by $M$, we investigate properties of the random closed 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 give conditions for the contact set $\mathcal{Z}$ to be countable or to have zero Lebesgue measure, and we obtain formulas that characterize the L\'evy measure of the associated subordinator. We study the limit of $\mathcal{Z}$ as $\alpha \to \infty$ and find for the so-called abrupt L\'evy processes introduced by Vigon that this limit is the set of local infima of $X$. When $X$ is a Brownian motion with drift $\beta$ such that $|\beta| < \alpha$, we calculate explicitly the densities of various random variables related to the minorant.