🤖 AI Summary
This study investigates the intrinsic connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces, with a focus on the relationship between covariance operators and positive definite functions. By introducing the γ-Radonifying operator, the authors characterize the covariance structure of weakly second-order Radon Gaussian measures in reproducing kernel Banach spaces and establish a rigorous functional-analytic foundation for sample paths of Gaussian processes. The main contributions include proving that the covariance operator is uniquely determined by a positive definite function, providing an operator-theoretic characterization specific to the Gaussian setting, and successfully extending the classical Driscoll’s theorem to the Banach space framework, thereby significantly broadening its applicability.
📝 Abstract
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.