Hölder regularity of a Wiener integral in abstract space

Abstract

In this article, we propose a way to consider processes indexed by a collection A of subsets of a general set T . A large class of vector spaces, manifolds and continuous 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 YA = A f dX for all A ∈ A for a deterministic function f : T → R and a set-indexed Lévy 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 Hölder 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 Balança and Herbin [6].