Homogeneous complex fractional Gaussian fields

Abstract

Where local boundary value problems are related to classical derivatives, non-local problems are stated in terms of fractional derivatives. We focus on the fractional Laplacian and its domain, which is a complex Hilbert space.

Gaussian fields on real Banach spaces are defined in a weak sense. We show that a standard Gaussian field on a Hilbert space cannot exist, but that an abstract Wiener space is a viable alternative. This concept turns out to be equivalent to that of a centred Gaussian field, and in some cases it may be constructed by way of a straightforward Hilbert-Schmidt operator.

This theory is repeated for the complex case, where identical results hold. By applying the theory of complex abstract Wiener spaces to the domain of the fractional Laplacian, we obtain the complex fractional Gaussian fields. In particular, we discuss the examples of the Gaussian free field and the bi-Laplacian field.