A note on the Malliavin-Sobolev spaces

TL;DR

This paper refines the understanding of Malliavin-Sobolev spaces by providing a strong formulation of stochastic Gâteaux differentiability and analyzing their internal set-inclusion structure.

Contribution

It introduces a strong formulation of stochastic Gâteaux differentiability and offers a new internal structure characterization of Malliavin-Sobolev spaces.

Findings

Established a strong formulation of stochastic Gâteaux differentiability.

Provided a new characterization of the internal structure of Malliavin-Sobolev spaces.

Analyzed the sharpness of a recent characterization of these spaces.

Abstract

In this paper, we provide a strong formulation of the stochastic G{\^a}teaux differentiability in order to study the sharpness of a new characterization, introduced in [6], of the Malliavin-Sobolev spaces. We also give a new internal structure of these spaces in the sense of sets inclusion.