Fredholm representation of multiparameter Gaussian processes with applications to equivalence in law and series expansions

TL;DR

This paper demonstrates that multiparameter Gaussian processes with integrable variance can be represented via Fredholm-type Wiener integrals, enabling analysis of their equivalence in law and series expansions.

Contribution

It introduces a Fredholm integral representation for multiparameter Gaussian processes and applies it to study process equivalence and construct series expansions.

Findings

Fredholm kernel as unique symmetric square root of covariance

Representation facilitates analysis of process equivalence in law

Allows construction of series expansions for Gaussian processes

Abstract

We show that every multiparameter Gaussian process with integrable variance function admits a Wiener integral representation of Fredholm type with respect to the Brownian sheet. The Fredholm kernel in the representation can be constructed as the unique symmetric square root of the covariance. We analyze the equivalence of multiparameter Gaussian processes by using the Fredholm representation and show how to construct series expansions for multiparameter Gaussian processes by using the Fredholm kernel.