On some laws of iterated logarithm for Burgers turbulence with Brownian initial data based on the concave majorant

We study the shock structure and the asymptotic behaviour of some flux across the origin in one-dimensional Burgers turbulence, the entropy solution to the inviscid Burgers equation, with random initial velocity for the uniformly distributed particles on the positive half line. We assume, in contrast to other works on Burgers turbulence, initially a vacuum state on the negative half line. We also obtain some asymptotic estimates for the concave majorant of Brownian motion.