I am trying to understand some examples of computing the spectral density of SPDEs. The root of my confusion is that some terms appear to be missing from the calculation of the squared transform, and I can't work out why.
Example 1
First, here is an example I can follow. For the SPDE
$(\kappa^2 + \nabla \cdot \nabla)^{\alpha/2} u(s) = \mathcal{W}(s)$
we have
$\mathbb{E}(|\kappa^2 + \nabla \cdot \nabla)^{\alpha/2} \hat{u}(s)|^2) = \mathbb{E}(|\hat{\mathcal{W}}(s)|^2)$
and the spectral density is given by
$f(\omega) = (2\pi (\kappa^2 + \omega^2)^{\alpha})^{-1}$.
Example 2
However, the next example I cannot understand. For the SPDE given by
$\tau (\varphi^2 - \frac{\partial^2}{\partial t^2})^{1/2} u(s) = \mathcal{W}(s)$
the spectral density is given as
$f(\omega) = (\tau 2\pi (\varphi^2 + \omega^2))^{-1}$
but why is it not $\tau^2$?
Example 3
Similarly, a later example considers the SPDE given by
$\tau [\rho \frac{\partial}{\partial t} - \Delta + \kappa^2] u(s,t) = \mathcal{W}(s,t)$
and the spectral density is given as
$f_{hd}(\lambda, \omega) = ((2\pi)^{d+1} \tau [\rho^2\omega^2 + (\kappa^2 + \lambda^2)^2])^{-1}$
whereas I would have expected to see more cross terms, e.g. $\rho^2\omega^2(\kappa^2 + \lambda^2)$.
For example 3, I had missed that the modulus was being taken over complex numbers. The power spectral density is given by
$S(\omega) = |\hat{f}(i \omega)^2| = \hat{f}(i \omega) \hat{f}(-i \omega)$
and in this particular case
$S(\rho \frac{\partial}{\partial t} - \nabla + \kappa^2) = (\rho i \omega + \lambda^2 + \kappa^2)(-\rho i \omega + \lambda^2 + \kappa^2) = \rho^2 \omega^2 + (\lambda^2 + \kappa^2)^2$
which gives the expected answer. I'm still unsure about the $\tau$ in example 2.