Hölder regularity for operator scaling stable random fields

We investigate the sample path regularity of operator scaling αα-stable random fields. Such fields were introduced in [H. Biermé, M.M. Meerschaert, H.P. Scheffler, Operator scaling stable random fields, Stochastic Process. Appl. 117 (3) (2007) 312–332.] as anisotropic generalizations of self-similar fields and satisfy the scaling property {X(cEx);x∈Rd}=(fdd){cHX(x);x∈Rd} where EE is a d×dd×d real matrix and H>0H>0. In the case of harmonizable operator scaling random fields, the sample paths are locally Hölderian and their Hölder regularity is characterized by the eigen decomposition of RdRd with respect to EE. In particular, the directional Hölder regularity may vary and is given by the eigenvalues of EE. In the case of moving average operator scaling αα-stable random fields, with α∈(0,2)α∈(0,2) and d≥2d≥2, the sample paths are almost surely discontinuous.