Some Families of Random Fields Related to Multiparameter Lévy Processes

Abstract

Let \({\mathbb {R}}^N_+= [0,\infty )^N\). We here make new contributions concerning 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 provide a Phillips formula concerning the composition of \((X_t)_{t\in {\mathbb {R}}^N_+}\) by means of subordinator fields. We finally define 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 present a model of anomalous diffusion in an anisotropic medium which extends the one treated in Beghin et al. (Stoch Proc Appl 130:6364–6387, 2020), by improving some of its shortcomings.