Excursions away from the Lipschitz minorant of a Levy process

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 Levy 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 Levy 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.