Exponential moment of a Gaussian linear functional

If a linear functional $Z=\int a(\mathbf{r})V(\mathbf{r})d\mathbf{r}$ of a real Gaussian random field exists in mean square, it is a Gaussian random variable. Completing the square in its density givesFor a complex linear functional $\varphi$, apply this to $Z=2\mathrm{Re}\varphi$; then $\frac{1}{2}\mathrm{Var}Z=\mathbb{E}|\varphi -\mathbb{E}\varphi {|}^2+\mathrm{Re}\mathbb{E}(\varphi -\mathbb{E}\varphi {)}^2$.

For a real standard Gaussian random vector $Z$, a real symmetric matrix $Q$, and $I-2Q$ strictly positive,Diagonalize $Q$ and complete the square in each coordinate. Failure of strict positivity makes the positive exponential integral divergent. The same formula extends to a self-adjoint trace-class quadratic form of an isonormal Gaussian process, with square-summable linear coefficients and a Fredholm determinant. A linear-plus-quadratic Gaussian random field functional is reduced to this form by its covariance square root.