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Complex abstract Wiener spaces

Tess J. van Leeuwen, Wioletta M. Ruszel

TL;DR

This work develops a comprehensive theory of $\mathbb{K}$-abstract Wiener spaces for $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ by recasting covariance in terms of a non-negative self-adjoint trace-class operator and using characteristic functions. It proves a fundamental equivalence between symmetric Gaussian fields and abstract Wiener spaces, and establishes a precise real–complex link via a bounded real structure, enabling a coherent transfer between $\mathbb{R}$-AWS and $\mathbb{C}$-AWS. The paper provides rigorous existence and uniqueness results, and demonstrates how complex Gaussian fields can be analyzed within the AWS framework, including a complex Feynman–Kac formula and complex fractional Gaussian fields. It also supplies concrete constructions of complex Brownian motion and complex FGFs, highlighting potential extensions to large deviations and Malliavin calculus in the complex setting. Overall, it unifies real and complex Gaussian analysis in infinite dimensions and expands the repertoire of complex stochastic models in probability and mathematical physics.

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.

Complex abstract Wiener spaces

TL;DR

This work develops a comprehensive theory of -abstract Wiener spaces for by recasting covariance in terms of a non-negative self-adjoint trace-class operator and using characteristic functions. It proves a fundamental equivalence between symmetric Gaussian fields and abstract Wiener spaces, and establishes a precise real–complex link via a bounded real structure, enabling a coherent transfer between -AWS and -AWS. The paper provides rigorous existence and uniqueness results, and demonstrates how complex Gaussian fields can be analyzed within the AWS framework, including a complex Feynman–Kac formula and complex fractional Gaussian fields. It also supplies concrete constructions of complex Brownian motion and complex FGFs, highlighting potential extensions to large deviations and Malliavin calculus in the complex setting. Overall, it unifies real and complex Gaussian analysis in infinite dimensions and expands the repertoire of complex stochastic models in probability and mathematical physics.

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 -AWS, where using the language of characteristic functions instead of measurable norms. In particular, we will prove that is a centred resp. proper -valued Gaussian field over 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 is equivalent to the -AWS. Finally we will relate the -AWS to the -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.

Paper Structure

This paper contains 14 sections, 20 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. K-random variables
  4. K-Gaussian fields
  5. Abstract Wiener spaces
  6. Real structures
  7. Results
  8. Proofs
  9. Proof of Theorem 3.1
  10. Proof of Theorem 3.2
  11. Proof of Theorem 3.3
  12. Examples
  13. The complex Wiener measure
  14. Complex fractional Gaussian fields

Key Result

Theorem 1

$X$ is a symmetric $H$-valued Gaussian field over $\mathbb{K}$ iff the covariance function can be written in terms of some non-negative, self-adjoint trace class operator.

Theorems & Definitions (44)

  • Theorem : Covariance as trace class operator
  • Theorem : Existence and uniqueness
  • Theorem
  • Example 2.1
  • Proposition 2.2: Proposition 24.2.11 in lapidoth
  • Definition 2.3
  • Example 2.4
  • Theorem 2.5: Fernique's Theorem
  • proof : Proof sketch
  • Remark 2.6
  • ...and 34 more