  Consider an inviscid Burgers equation whose initial data is a Levy a-stable
process Z with a > 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/a, as soon as a is close to 1. This gives a negative
answer to a conjecture of Janicki and Woyczynski. Along the way, we contradict
a recent conjecture of Z. Shi about the lower tails of integrated stable
processes.
