Stationary spectrum of a linear fluctuating interface ($⟨h_q^2⟩={\sigma }^2/(2\alpha q^2)$)

For a nonzero Fourier mode satisfying $d{h}_q=-\alpha q^2{h}_{q}dt+\sigma d{W}_q$ with $\alpha >0$, the explicit Ornstein-Uhlenbeck solution and Itô isometry give the stationary second moment ${\sigma }^2/(2\alpha q^2)$. The prefactor depends on the normalization of the mode noise, while the inverse-square dependence follows from diffusive relaxation.