# Excursions away from the Lipschitz minorant of a L\'evy process

## TL;DR

This paper characterizes the excursions of a Lévy process away from its Lipschitz minorant, providing a detailed description of the contact set and excursions, especially for Brownian motion with drift.

## Contribution

It offers a novel analysis of the contact set and excursions of Lévy processes relative to their Lipschitz minorant, extending excursion theory to this context.

## Key findings

- Conditions for the existence of the Lipschitz minorant of Lévy processes.
- Description of the contact set as a stationary regenerative set.
- Explicit path decompositions for Brownian motion with drift.

## Abstract

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.