I am looking for a reference of the following theorem
A Borel probability measure $\mu$ on a topological vector space $X$ is centered Gaussian if and only if there exists a positive semidefinite symmetric bilinear form $q: X^{\ast} \times X^{\ast} \rightarrow \mathbb{R}$ s.t.
$$ \int_X \exp(i f(x)) ~ d \mu(x) = \exp(- \frac{1}{2} ~q(f,f))$$
One direction can be shown as follows:
Assume $\mu$ is Gaussian, then by definition every $f \in X^{\ast}$ is a Gaussian random variable $X \rightarrow \mathbb{R}$. Hence
\begin{equation} \int_X e^{if(x)} d \mu(x) = \int_{\mathbb{R}} e^{iy} d \underbrace{f_{\ast}\mu(y)}_{\sim N(0, \text{Var}(f))} = \varphi_f(1) = \exp(- \frac{1}{2} \text{Var}(f)) \end{equation}
In finite dimension, the other dimension is done via Fourier inversion, but I am not sure about the infinite case.