Concave Majorant of Stochastic Processes and Burgers Turbulence

The asymptotic solution of the inviscid Burgers equations with initial potential ψ is closely related to the convex hull of the graph of ψ. In this paper, we study this convex hull, and more precisely its extremal points, if ψ is a stochastic process. The times where those extremal points are reached, called extremal times, form a negligible set for Lévy processes, their integrated processes, and Itô processes. We examine more closely the case of a Lévy process with bounded variation. Its extremal points are almost surely countable, with accumulation only around the extremal values. These results are derived from the general study of the extremal times of ψ+f, where ψ is a Lévy process and f a smooth deterministic drift. These results allow us to show that, for an inviscid Burgers turbulence with a compactly supported initial potential ψ, the only point capable of being Lagrangian regular is the time T where ψ reaches its maximum, and that is indeed a regular point if and only if 0 is regular for both half-lines. As a consequence, if the turbulence occurs on a non-compact interval, there are almost surely no Lagrangian regular points.