Random fields and the geometry of Wiener space

TL;DR

This paper extends Gaussian geometric functionals to infinite dimensions to analyze the geometry of smooth random fields within Wiener space, providing new tools for understanding their probabilistic and geometric properties.

Contribution

It introduces infinite dimensional extensions of Gaussian Minkowski functionals and applies them to study the geometry of smooth Wiener functionals.

Findings

Extended Gaussian Minkowski functionals to infinite dimensions

Analyzed geometric properties of smooth Wiener functionals

Provided new methods for probabilistic geometric analysis

Abstract

In this work we consider infinite dimensional extensions of some finite dimensional Gaussian geometric functionals called the Gaussian Minkowski functionals. These functionals appear as coefficients in the probability content of a tube around a convex set $D\subset\mathbb{R}^k$ under the standard Gaussian law $N(0,I_{k\times k})$. Using these infinite dimensional extensions, we consider geometric properties of some smooth random fields in the spirit of [Random Fields and Geometry (2007) Springer] that can be expressed in terms of reasonably smooth Wiener functionals.