On the Hausdorff Dimension of Regular Points of Inviscid Burgers Equation with Stable Initial Data

Consider an inviscid Burgers equation whose initial data is a Lévy α−stable process Z with α > 1. We show that when Z has positive jumps, the Hausdorff dimension of the set of Lagrangian regular points associated with the equation is strictly smaller than 1/α, as soon as α is close to 1. This gives a negative answer to a conjecture of Janicki and Woyczynski [13]. Along the way, we contradict a recent conjecture of Z. Shi about the lower tails of integrated stable processes.