New classes of spectral densities for lattice processes and random fields built from simple univariate margins

Emilio Porcu,Martin Ostoja-Starzewski,Pablo Gregori,Jorge Mateu

doi:10.1007/s00477-012-0572-2

Abstract

AbstractQuasi arithmetic and Archimedean functionals are used to build new classes of spectral densities for processes defined on anyd-dimensional lattice$${\mathbb{Z}^d}$$and random fields defined on thed-dimensional Euclidean space$${\mathbb{R}^d}$$, given simple margins. We discuss the mathematical features of the proposed constructions, and show rigorously as well as through examples, that these new classes of spectra generalize celebrated classes introduced in the literature. Additionally, we obtain permissible spectral densities as linear combinations of quasi arithmetic or Archimedean functionals, whose associated correlation functions may attain negative values or oscillate between positive and negative ones. We finally show that these new classes of spectral densities can be used for nonseparable processes that are not necessarily diagonally symmetric.

Highlights

Modern mechanics of materials is largely driven by multiscale problems
Quasi arithmetic and Archimedean functionals are used to build new classes of spectral densities for processes defined on any d-dimensional lattice Zd and random fields defined on the d-dimensional Euclidean space Rd, given simple margins
The representative volume element (RVE) should be replaced by a statistical volume element (SVE) and a deterministic field of material properties is replaced by a random field (RF)

Summary

Introduction

Modern mechanics of materials is largely driven by multiscale problems. The label under which much is done is called homogenization, where a deterministic approach is tacitly adopted. We propose new classes of nonseparable spectral densities for either processes defined on infinite regular lattices of Zd or RFs defined on Rd. We shall show that our approach allows for building nonseparable spectral structures starting from simple margins, and sometimes permits to specify the corresponding correlation function, the associated stochastic representation is all but trivial. 1930) and Archimedean functionals and use them to build new models of nonseparable spectral densities defined on any d-dimensional regular lattice or Euclidean space, given a number of margins We show that this procedure allows for a wide class of spectral densities that admit as a special case the separable one.

Preliminaries

Axiomatica

The lattice case

The RF case

Conclusions