Let $E$ be a separable Banach space, consider a centered $E$-valued Gaussian process $\{x_t,t\ge 0\}$ that satisfies \begin{equation} \mathbb{E}\phi(x_s)\psi(x_t)=R(s,t)K(\phi,\psi),\quad \phi,\psi\in E^*, s,t\ge 0, \end{equation} where $R,K$ are some (covariance) functions.
I'm trying to find some literature about such processes. This decomposition itself is quite common, the most notable example being an $E$-valued Brownian motion, in which case $R(s,t)=s\wedge t$, and $K$ can be the covariance function of an arbitrary Gaussian measure on $E$. Has there been any study on the properties of such processes in general case?
