Harmonizable Multifractional Stable Field: Sharp results on sample path behavior

Abstract

For about three decades now, there is an increasing interest in study of multifractional processes/fields. The paradigmatic example of them is Multifractional Brownian Field (MBF) over $R^N$, which is a Gaussian generalization with varying Hurst parameter (the Hurst function) of the well-known Fractional Brownian Motion (FBM). Harmonizable Multifractional Stable Field (HMSF) is a very natural (and maybe the most natural) extension of MBF to the framework of heavy-tailed Symmetric $\alpha$-Stable (S$\alpha$S) distributions. Many methods related with Gaussian fields fail to work in such a non-Gaussian framework, this is what makes study of HMSF to be difficult. In our article we construct wavelet type random series representations for the S$\alpha$S stochastic field generating HMSF and for related fields. Then, under weakened versions of the usual Hölder condition on the Hurst function, we obtain sharp results on sample path behavior of HMSF: optimal global and pointwise moduli of continuity, quasi-optimal pointwise modulus of continuity on a universal event of probability 1 not depending on the location, and an estimate of the behavior at infinity which is optimal when the Hurst function has a limit at infinity to which it converges at a logarithmic rate.