Rough paths analysis of general Banach space-valued Wiener processes

Abstract

In this article, we carry out a rough paths analysis for Banach space-valued Wiener processes. We show that most of the features of the classical Wiener process pertain to its rough path analog. To be more precise, the enhanced process has the same scaling properties and it satisfies a Fernique type theorem, a support theorem and a large deviation principle in the same Hölder topologies as the classical Wiener process does. Moreover, the canonical rough paths of finite dimensional approximating Wiener processes converge to the enhanced Wiener process. Finally, a new criterion for the existence of the enhanced Wiener process is provided which is based on compact embeddings. This criterion is particularly handy when analyzing Kunita flows by means of rough paths analysis which is the topic of a forthcoming article.