On the Hausdorff dimension of regular points of inviscid Burgers equation with stable initial data

Abstract

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.

References in corpus (3)

• The law of the supremum of a stable Lévy process with no negative jumps
• Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction
• The lower tail problem for homogeneous functionals of stable processes with no negative jumps

Cited by in corpus (3)

• Persistence probabilities \& exponents
• The lower tail problem for homogeneous functionals of stable processes with no negative jumps
• Persistence of integrated stable processes