Complex abstract Wiener spaces

Abstract

Real abstract Wiener spaces (AWS) were originally defined by Gross using measurable norms, as a generalisation of the theory of advanced integral calculus in infinite dimensions as introduced by Cameron and Martin. In this paper we present a rigorous, complete and self-contained general framework for $\mathbb{K}$-AWS, where $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$ using the language of characteristic functions instead of measurable norms. In particular, we will prove that $X$ is a centred resp. proper $H$-valued Gaussian field over $\mathbb{K}$ iff the covariance function can be written in terms of some non-negative, self-adjoint trace class operator, and that the existence and uniqueness of $X$ is equivalent to the $\mathbb{K}$-AWS. Finally we will relate the $\mathbb{C}$-AWS to the $\mathbb{R}$-AWS by way of a real structure, which is a real linear, complex anti-linear involution on a complex vector space. This allows for a commutative relation between the real and complex Gaussian fields and the real and complex abstract Wiener spaces. We will construct specific examples which fall under this framework like the complex Brownian motion, complex Feynman-Kac formula and complex fractional Gaussian fields.