ABSTRACT WIENER PROCESSES AND THEIR REPRODUCING KERNEL HILBERT SPACES.

The paper explores the relationship between Gaussian processes and their associated RKH Spaces. A simple proof of Grosss theorem on abstract Wiener spaces is given. For a Gaussian measure mu with continuous covariance R defined on the Banach space CT of real continuous functions on T T being a separable complete metric space it is shown that the closure of HR in CT is the support of mu. This result is extended to Gaussian measures on arbitrary separable Banach spaces. A necessary and sufficient criterion for a separable Gaussian process xt 0 or t or 1 with continuous covariance R to have continuous sample paths is furnished by the following result to the effect that the canonical normal distribution on HR extends to a Gaussian measure on C0,1 if and only if the sup-norm on HR is measurable in the sense of Gross. Author