Can realizations of a filtered Gaussian white noise process be represented as a Fourier transform?

Suppose we have a noise process $V(t)$ which is the result of passing Gaussian white noise through a filter with frequency response function $H(\omega)$. Can we represent realizations of this process as a Fourier transform $$V(t) = \int \frac{d\omega}{2\pi} M(\omega) e^{i \phi(\omega)}$$ where $M(\omega)$ and $\phi(\omega)$ are random variables? If so, what are the statistics of $M(\omega)$ and $\phi(\omega)$?