A strange definition about reproducing kernel Hilbert spaces for symmetric Gaussian measures on a separable Banach space

Question

In stochastic equations in infinite dimensions written by Da Prato reproducing kernel Hilbert space is defined

Let $\mu$ be a symmetric Gaussian measures on a separable Banach space $E$. A linear subspace $H \subset E$ equipped with a Hilbert norm $|\cdot|_{H}$ is said to be a reproducing kernel Hilbert space for $\mu$ if H is complete, continuously embedded in $E$ and such that for arbitrary $\varphi \in E^{\star}$, the distribution of $\varphi$ is Gaussian $\mathcal{N}(0,|\varphi|^2_{H})$, where $$|\varphi|_{H}=\sup_{|h|\leq 1, h\in H}|\varphi(h)|$$

My question is

(1) How can this definition coincide with the general definition such as

http://en.wikipedia.org/wiki/RKHS

(2) What is the advantage of this defination? Is there any convenience?

Answer 1

The two notions coincide in the particular case when $E = C(X)$ consists of the continuous functions from $X$ to $\mathbf{R}$ with $X$ playing the same role as in the Wikipedia article. (Say $X$ is a compact space to avoid technicalities.) In this case, if $H$ is defined as in Da Prato, since $H \subset C(X)$, the point evaluations $L_x$ are indeed continuous, so you are in the setting of the Wikipedia article. The "reproducing kernel" $K$ is then nothing but the covariance function of the measure $\mu$. In other words, the definition given by Da Prato is precisely the one guaranteeing that the reproducing kernel $K$ is given by $$ K(x,y) = \int_E f(x)f(y) \mu(df)\;. $$ It is true that the terminology is a bit confusing, many authors prefer to call $H$ the Cameron-Martin space of $\mu$ instead.