Iterated Integrals of Finite- and Infinite-Dimensional Gaussian Processes

Abstract

In this paper, we find conditions under which the known definitions of iterated integrals of finite-dimensional Gaussian processes could be used to construct integrals of the infinite-dimensional analogues of these processes. In the context of the rough paths theory, our results allow us to carry the existing constructions of finite-dimensional Gaussian rough paths over to the infinite-dimensional case. Using more abstract terms, we define a continuous linear map between the space of Gaussian chaoses generated by some Gaussian processes and the space of chaoses generated by the infinite-dimensional analogues of these processes.