Rough path lifts of Banach space-valued Gaussian processes

Abstract

Under certain assumptions on a Gaussian process taking values in a separable Banach space, we construct its lift to a geometric rough path. The lift is natural in the sense that for any sequence of piece-wise linear approximations to the original process, their signatures converge to the lifted path in a suitable metric. This extends to infinite dimensions the known results in Euclidean spaces. Examples of processes satisfying our conditions include the infinite-dimensional analogues of Brownian motion, fractional Brownian motion with Hurst parameter $H\in (\frac{1}{4},\frac{1}{2}]$, Ornstein–Uhlenbeck process. As a by-product of our methods, we also provide a construction for Ito–Skorokhod integrals of these processes, which might be of independent interest.