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

Tommi Sottinen,Lauri Viitasaari

doi:10.15559/15-vmsta39cnf

Tommi Sottinen, Lauri Viitasaari

Open Access

https://doi.org/10.15559/15-vmsta39cnf

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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.

Highlights

The Brownian sheet can be considered as the Gaussian white noise on [0, 1] with the Lebesgue control measure. This means that dW is a random measure on ([0, 1], B([0, 1]), Leb([0, 1])) characterized by the following properties: 1. A dWt ∼ N (0, Leb(A)), 2
We show the Fredholm representation for Gaussian fields satisfying the trace condition (3) in Section 2, Theorem 1
We do not do that in this article, it would be quite straightforward given the results for the one-dimensional case provided in [9]

Summary

Introduction

We consider multiparameter processes, that is, our time is multidimensional. We use the following notation throughout this article: t, s, u ∈ Rn are n-dimensional multiparameters of time: t = The Brownian sheet can be considered as the Gaussian white noise on [0, 1] with the Lebesgue control measure. This means that dW is a random measure on ([0, 1], B([0, 1]), Leb([0, 1])) characterized by the following properties: 1. We show the Fredholm representation for Gaussian fields satisfying the trace condition (3) in Section 2, Theorem 1. The Fredholm representation of Theorem 1 can be used to provide a transfer principle that builds stochastic analysis and Malliavin calculus for Gaussian fields from the corresponding well-known theory for the Brownian sheet. We do not do that in this article, it would be quite straightforward given the results for the one-dimensional case provided in [9]

Fredholm representation

Application to equivalence in law

Application to series expansions