Some families of random fields related to multiparameter Lévy processes

Let ${\mathbb{R}}_{+}^N=[0,\mathrm{\infty }{)}^N$. We here consider a class of random fields $(X_t{)}_{t\in {\mathbb{R}}_{+}^N}$ which are known as Multiparameter Lévy processes. Related multiparameter semigroups of operators and their generators are represented as pseudo-differential operators. We also consider the composition of $(X_t{)}_{t\in {\mathbb{R}}_{+}^N}$ by means of the so-called subordinator fields and we provide a Phillips formula. We finally study the composition of $(X_t{)}_{t\in {\mathbb{R}}_{+}^N}$ by means of the so-called inverse random fields, which gives rise to interesting long range dependence properties. As a byproduct of our analysis, we study a model of anomalous diffusion in an anisotropic medium which extends the one treated in [8].