I am working with the following SDE: $$\dot{x}= u'(x)+\sqrt{2D} \hspace{1mm} \xi(t)$$
where $u(x)=- \beta|x|$ and the initial conditions is $x=\delta({x-x_0})$ and $\xi(t)$ is a white Gaussian noise.
This process will have a stationary solution at the end, namely: $p_0(x)=\frac{D}{2 \beta}exp(-\frac{\beta}{D} |x|)$
(we get this stationary solution by transforming this SDE into a Fokker-Planck equation.)
So my question is, how could I solve this SDE? How could I know whether an analytical solution exists? A very similar equations but with other boundary conditions have solution here, and I was wondering if an analytical solution could also exist for this one.
Any tips or suggested readings would be highly really appreciated.