Lipschitz minorants of Brownian motion and L,vy processes

For , the -Lipschitz minorant of a function is the greatest function such that and for all , should such a function exist. If is a real-valued L,vy process that is not pure linear drift with slope , then the sample paths of have an -Lipschitz minorant almost surely if and only if . Denoting the minorant by , we investigate properties of the random closed set , which, since it is regenerative and stationary, has the distribution of the closed range of some subordinator "made stationary" in a suitable sense. We give conditions for the contact set to be countable or to have zero Lebesgue measure, and we obtain formulas that characterize the L,vy measure of the associated subordinator. We study the limit of as and find for the so-called abrupt L,vy processes introduced by Vigon that this limit is the set of local infima of . When is a Brownian motion with drift such that , we calculate explicitly the densities of various random variables related to the minorant.