Some families of random fields related to multiparameter L\'evy processes

TL;DR

This paper explores a class of multiparameter Lévy-based random fields, their operator representations, compositions via subordinator and inverse fields, and applications to anomalous diffusion in anisotropic media.

Contribution

It introduces a framework for analyzing multiparameter Lévy processes, including their operator forms, composition methods, and implications for anomalous diffusion models.

Findings

Representation of semigroups as pseudo-differential operators

Development of a Phillips formula for subordinator compositions

Identification of long-range dependence in inverse field compositions

Abstract

Let $\mathbb{R}^N_+= [0,\infty)^N$. We here consider a class of random fields $(X_t)_{t\in \mathbb{R}^N_+}$ which are known as Multiparameter L\'evy 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].