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 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

Survival probability of the branching random walk killed below a linear boundary

We give an alternative proof of a result by N. Gantert, Y. Hu and Z. Shi on the asymptotic behavior of the survival probability of the branching random walk killed below a linear boundary, in the special case of deterministic binary branching and bounded random walk steps. Connections with the Brunet-Derrida theory of stochastic fronts are discussed

Persistence of integrated stable processes

We compute the persistence exponent of the integral of a stable L\'evy process in terms of its self-similarity and positivity parameters. This solves a problem raised by Z. Shi (2003). Along the way, we investigate the law of the stable process L evaluated at the first time its integral X hits zero, when the bivariate process (X,L) starts from a coordinate axis. This extends classical formulae by McKean (1963) and Gor'kov (1975) for integrated Brownian motion