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

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

2007-02-09

arxiv.org/abs/math/0702260v2

Mathematics

Probability

Scientific paper

10.1007/s10955-008-9508-0

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.

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