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].
Categories: math.PR