Mutual information in Gaussian channels

In the Gaussian channel Y(t) = Φ(t) + X(t) = message + noise, where Φ(t) and X(t) are mutually independent, the information I(Y, Φ) is evaluated. One of the results is that I(Y, Φ) < ∞ if and only if Φ ? $\text{H}$(X) = the reproducing kernel Hilbert space for X(·). And the causal formula of I(Y, Φ) is given.