Hausdorff, Large Deviation and Legendre Multifractal Spectra of L\'evy Multistable Processes

We compute the Hausdorff multifractal spectrum of two versions of multistable L{'e}vy motions. These processes extend classical L{'e}vy motion by letting the stability exponent $\alpha$ evolve in time. The spectra provide a decomposition of [0, 1] into an uncountable disjoint union of sets with Hausdorff dimension one. We also compute the increments-based large deviations multifractal spectrum of the independent in-crements multistable L{'e}vy motion. This spectrum turns out to be concave and thus coincides with the Legendre multifractal spectrum, but it is different from the Haus-dorff multifractal spectrum. The independent increments multistable L{'e}vy motion thus provides an example where the strong multifractal formalism does not hold.