Brownian zero mode of a fluctuating interface ($d{h}_0=\sigma d{W}_t$)

An unpinned uniform height driven by Gaussian white noise is Brownian motion. Its variance grows as ${\sigma }^{2}t$ and it has no stationary distribution on the real height axis for $\sigma >0$. Indeed, a stationary characteristic function would obey $\hat{\mu }(k)=\hat{\mu }(k)e^{-{\sigma }^2{k}^{2}t/2}$, incompatible with continuity at zero. Quotienting out the uniform height can still leave stationary shape fluctuations.