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Concave majorant of stochastic processes and structure of shocks in Burgers turbulence

In this paper, we study the least concave majorant of a stochastic process, and more precisely the set of times where its slope varies, which we call the extremal superior times. We establish the negligibility of this set, namely the extremal superior set, for certain classes of processes, including Lévy processes, their integrated processes, and Itô processes. We examine more closely the case of a Lévy process of bounded variation. We show that when its drift is null, its extremal superior set is almost surely countable, with accumulation only around the point where X tends to its maximum value. When a parabolic concave drift is added to the initial process, the extremal superior set models a one-dimensional fluid ruled by Burgers equation with vanishing viscosity. We establish results about the extremal superior set of a bounded variation Lévy process to which a sufficiently smooth drift is added, and derive results about the shock structure of the fluid, with particular interest in the set of Lagrangian regular points.