Structure of shocks in Burgers turbulence with L\'evy noise initial data

We study the structure of the shocks for the inviscid Burgers equation in dimension 1 when the initial velocity is given by L\'evy noise, or equivalently when the initial potential is a two-sided L\'evy process $\psi_0$. When $\psi_0$ is abrupt in the sense of Vigon or has bounded variation with $\limsup_{|h| \downarrow 0} h^{-2} \psi_0(h) = \infty$, we prove that the set of points with zero velocity is regenerative, and that in the latter case this set is equal to the set of Lagrangian regular points, which is non-empty. When $\psi_0$ is abrupt we show that the shock structure is discrete. When $\psi_0$ is eroded we show that there are no rarefaction intervals.