Some Families of Random Fields Related to Multiparameter Lévy Processes

Abstract

AbstractLet $${\mathbb {R}}^N_+= [0,\infty )^N$$ R + N = [ 0 , ∞ ) N . We here make new contributions concerning a class of random fields $$(X_t)_{t\in {\mathbb {R}}^N_+}$$ ( X t ) t ∈ 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 $$(X_t)_{t\in {\mathbb {R}}^N_+}$$ ( X t ) t ∈ R + N by means of subordinator fields. We finally define the composition of $$(X_t)_{t\in {\mathbb {R}}^N_+}$$ ( X t ) t ∈ 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.