Concave majorant of stochastic processes and Burgers turbulence

Abstract

The asymptotic solution of the inviscid Burgers equations with initial potential $\psi$ is closely related to the convex hull of the graph of $\psi$. In this paper, we study this convex hull, and more precisely its extremal points, if $\psi$ is a stochastic process. The times where those extremal points are reached, called extremal times, form a negligible set for L\'evy processes, their integrated processes, and It\^o processes. We examine more closely the case of a L\'evy 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 $\psi+f$, where $\psi$ is a L\'evy 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 $\psi$, the only point capable of being Lagrangian regular is the time $T$ where $\psi$ reaches its maximum, and that is indeed a regular point iff 0 is regular for both half-lines. As a consequence, if the turbulence occurs on a non-compact interval, there are a.s. no Lagrangian regular points.