Two continua of embedded regenerative sets (2003.05009v1)

Abstract: Given a two-sided real-valued L\'evy process $(X_t){t \in \mathbb{R}}$, define processes $(L_t){t \in \mathbb{R}}$ and $(M_t){t \in \mathbb{R}}$ by $L_t := \sup{h \in \mathbb{R} : h - \alpha(t-s) \le X_s \text{ for all } s \le t} = \inf{X_s + \alpha(t-s) : s \le t}$, $t \in \mathbb{R}$, and $M_t := \sup { h \in \mathbb{R} : h - \alpha|t-s| \leq X_s \text{ for all } s \in \mathbb{R} } = \inf {X_s + \alpha |t-s| : s \in \mathbb{R}}$, $t \in \mathbb{R}$. The corresponding contact sets are the random sets $\mathcal{H}\alpha := { t \in \mathbb{R} : X_{t}\wedge X_{t-} = L_t}$ and $\mathcal{Z}\alpha := { t \in \mathbb{R} : X{t}\wedge X_{t-} = M_t}$. For a fixed $\alpha>\mathbb{E}[X_1]$ (resp. $\alpha>|\mathbb{E}[X_1]|$) the set $\mathcal{H}\alpha$ (resp. $\mathcal{Z}\alpha$) is non-empty, closed, unbounded above and below, stationary, and regenerative. The collections $(\mathcal{H}{\alpha}){\alpha > \mathbb{E}[X_1]}$ and $(\mathcal{Z}{\alpha}){\alpha > |\mathbb{E}[X_1]|}$ are increasing in $\alpha$ and the regeneration property is compatible with these inclusions in that each family is a continuum of embedded regenerative sets in the sense of Bertoin. We show that $(\sup{t < 0 : t \in \mathcal{H}\alpha}){\alpha > \mathbb{E}[X_1]}$ is a c`adl`ag, nondecreasing, pure jump process with independent increments and determine the intensity measure of the associated Poisson process of jumps. We obtain a similar result for $(\sup{t < 0 : t \in \mathcal{Z}\alpha}){\alpha > |\beta|}$ when $(X_t)_{t \in \mathbb{R}}$ is a (two-sided) Brownian motion with drift $\beta$.