The Hausdorff dimension of the range of the Lévy multistable processes

Abstract

We compute the Hausdorff dimension of the image X(E) of a non random Borel set E $\subset$ [0, 1], where X is a Lévy multistable process in R. This extends the case where X is a classical stable Lévy process by letting the stability exponent $\alpha$ be a smooth function, which leads to non-homogeneous processes because their increments are not stationary and not necessarily independent. Contrary to the situation where the stability parameter is a constant, the dimension depends on the version of the multistable Lévy motion when the process has an infinite first moment.