Gaussian Processes with Sample Paths in Reproducing Kernel Banach Spaces

We investigate the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces. We show that the covariance operator of a weak second-order Radon probability measure on such a space is uniquely determined by a positive definite function. In the Gaussian case, we characterize those positive definite functions that arise from covariance operators in terms of γ-radonifying operators. Building on these results, we extend the classical Driscoll theorem to the Banach space setting.