= Solution
The <Neumann boundary conditions> give <Fourier modes> $q=n\pi/L$, including the spatially uniform mode $q=0$. In the frame translating with the mean deposition height, that mode has no restoring force and obeys
$$
dh_0=\sigma\,dW_t,\qquad
h_0(t)=h_0(0)+\sigma W_t,
\qquad \boxed{\operatorname{Var}h_0(t)=\operatorname{Var}h_0(0)+\sigma^2t}.
$$
Here $W_t$ is <Brownian motion>, independent of the initial height. More generally $\langle h_0(t)^2\rangle=\langle h_0(0)^2\rangle+\sigma^2t$ when these <second moments> exist. Removing the mean deposition drift does not remove the <Gaussian white noise>.
The <Brownian zero mode of a fluctuating interface> has no <stationary distribution> on the real height axis for $\sigma^2>0$. This conclusion does not depend on assuming finite <variance>: a stationary <characteristic function> would satisfy $\widehat\mu(k)=\widehat\mu(k)e^{-\sigma^2k^2t/2}$, hence vanish for every $k\ne0$, contradicting its continuity at zero and $\widehat\mu(0)=1$. Any stationary joint distribution of the whole height would have a stationary zero-mode marginal, which is impossible. Thus \b[the full unpinned height has no Boltzmann equilibrium]. Pinning the mean height, or retaining only the nonzero <Fourier modes>, removes this obstruction. The PDF has $\dot h_q$; the TeX's $\hbar_q$ is a transcription error.
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