Given a two-sided real-valued Lévy process (Xt)t∈R, define processes (Lt)t∈R and (Mt)t∈R by Lt:=sup{h∈R:h−α(t−s)≤Xs for all s≤t}=inf{Xs+α(t−s):s≤t}, t∈R, and Mt:=sup{h∈R:h−α∣t−s∣≤Xs for all s∈R}=inf{Xs+α∣t−s∣:s∈R}, t∈R. The corresponding contact sets are the random sets Hα:={t∈R:Xt∧Xt−=Lt} and Zα:={t∈R:Xt∧Xt−=Mt}. For a fixed α>E[X1] (resp. α>∣E[X1]∣) the set Hα (resp. Zα) is non-empty, closed, unbounded above and below, stationary, and regenerative. The collections (Hα)α>E[X1] and (Zα)α>∣E[X1]∣ are increasing in α 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∈Hα})α>E[X1] is a càdlàg, 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∈Zα})α>∣β∣ when (Xt)t∈R is a (two-sided) Brownian motion with drift β.
