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

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].