  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.
