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For $q\ne 0$ and $\alpha >0$, each Fourier mode is an Ornstein-Uhlenbeck process with decay rate $r_q=\alpha q^2$. Its explicit Ornstein-Uhlenbeck solution isso the Itô isometry givesTo identify the Overdamped Langevin dynamics, choose a friction ${\zeta }_q>0$, a harmonic oscillator potential $V_q={\zeta }_q\alpha q^2{h}_q^2/2$, and force noise $f_q={\zeta }_q{\eta }_q$. The fluctuation-dissipation relation for a Langevin particle then defines $k_B{T}_{\mathrm{eff}}={\zeta }_q{\sigma }^2/2$. Its Boltzmann distribution isThis is the stationary spectrum of a linear fluctuating interface, also following from the equipartition theorem. The numerical prefactor uses exactly the mode-noise normalization given in the paper. The nonzero modes can therefore have a stationary Gaussian distribution even though the unpinned zero mode cannot.

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.