Two continua of embedded regenerative sets
Abstract
Given a two-sided real-valued L\'evy process (Xt)t ∈ R, define processes (Lt)t ∈ R and (Mt)t ∈ R by Lt := \h ∈ R : h - α(t-s) Xs for all s t\ = ∈f\Xs + α(t-s) : s t\, t ∈ R, and Mt := \ h ∈ R : h - α|t-s| ≤ Xs for all s ∈ R \ = ∈f \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 (\t < 0 : t ∈ Hα\)α > E[X1] 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 (\t < 0 : t ∈ Zα\)α > |β| when (Xt)t ∈ R is a (two-sided) Brownian motion with drift β.