An improved characterisation of regular generalised functions of white noise and an application to singular SPDEs

TL;DR

This paper provides a new characterization of certain spaces of generalized functions related to white noise using U-functionals, and applies this to analyze singular SPDEs like stochastic transport and heat equations.

Contribution

It introduces a novel characterization of the spaces G_K and G_K' via U-functionals, simplifying the analysis of singular SPDEs.

Findings

New characterization of G_K and G_K' spaces.

Application to stochastic transport and heat equations with multiplicative noise.

Enhanced understanding of regularized generalized functions in SPDEs.

Abstract

A characterisation of the spaces $\mathcal{G}_K$ and $\mathcal{G}_K'$ introduced in Grothaus et al. (Methods Funct Anal Topol 3(2):46-64, 1997) and Potthoff and Timpel (Potential Anal 4(6):637-654, 1995) is given. A first characterisation of these spaces provided in Grothaus et al. (Methods Funct Anal Topol 3(2):46-64, 1997) uses the concepts of holomorphy on infinite dimensional spaces. We, instead, give a characterisation in terms of U-functionals, i.e., classic holomorphic function on the one dimensional field of complex numbers. We apply our new characterisation to derive new results concerning a stochastic transport equation and the stochastic heat equation with multiplicative noise.