see e.g. [@Nattermann1992_PRA]. The evolution equation of the Fourier modes has the structure of an Ornstein-Uhlenbeck process; its solutions are Gaussian random variables with the correlation function $$\langle A(q,t) A(q',t') \rangle = \frac{k_B T}{ {\cal T}} \frac{1}{q^2} \left(1 - \exp\left[ - \frac{2 {\cal T}}{B} q^2 t\right ] \right) \delta(q - q')$$ or in terms of the wave number $$\langle A(n,t) A(n',t') \rangle = \frac{2 k_B T}{ {\cal T} L} \left(\frac{L}{\pi n}\right)^2 \left(1 - 
\exp\left[ - \frac{2 \pi^2 {\cal T}}{BL^2} n^2 t\right ] \right) \delta_{nn'}$$

Spatially fluctuating forces
----------------------------

If the elastic line is subject to spatially fluctuating but temporally fixed forces, the evolution is given by the quenched Edwards-Wilkinson equation $$B \frac{d y}{d t} = {\cal T}\frac{\partial^2 y}{\partial x^2} + f_{\rm q} (x,y) + f_{\rm ext}
\label{eq:qedwilk}$$ where $f_{\rm ext}$ is an externally applied driving force per unit length and the random force $f_{\rm q}$ has the correlation function $$\langle f_{\rm q}(x,y) f_{\rm q}(x',y') \rangle = \hat{f}^2 \xi \delta(x-x') \phi(y-y').$$ where $\hat{f}$ is a characteristic force per unit length created by a random potential of correlation length $\xi$ and it is understood that we consider the line on scales well above the correlation length. The function $\phi(y)$ where $\phi(0) = 