H\"older regularity of a Wiener integral in abstract space

TL;DR

This paper introduces a framework for defining Wiener-type integrals over set-indexed processes in abstract spaces, providing new regularity bounds that relate the smoothness of the integrand and the process.

Contribution

It offers a simplified construction of set-indexed Wiener integrals in abstract spaces and derives new bounds for their Hölder regularity based on the properties of the integrand and the process.

Findings

Provided bounds for the Hölder regularity of the Wiener integral

Extended the setting to include vector spaces, manifolds, and trees

Demonstrated how regularities of the integrand and process influence the integral's smoothness

Abstract

In this article, we propose a way to consider processes indexed by a collection $\mathcal{A}$ of subsets of a general set $\mathcal{T}$. A large class of vector spaces, manifolds and continuous $\mathbb{R}$-trees are particular cases. Lattice-theoretic and topological assumptions are considered separately with a view to clarifying the exposition. We then define a Wiener-type integral $Y_A = \int_A f\,\text dX$ for all $A\in\mathcal{A}$ for a deterministic function $f:\mathcal{T} \rightarrow \mathbb{R}$ and a set-indexed L\'evy process $X$. It is a particular case of Raput and Rosinski [40], but our setting enables a quicker construction and yields more properties about the sample paths of $Y.$ Finally, bounds for the H\"older regularity of $Y$ are given which indicate how the regularities of $f$ and $X$ contributes to that of $Y$. This follows the works of Jaffard [24] and Balan\c{c}a and Herbin [6].