Sum of two random noises
Is the PSD of u1 and u2 computable ? Does it only depends on p1 and p2 ?

Yes. The Fourier Transform is a linear operation so the spectrum of the sum is simply the sum of the spectra. The PSD is just magnitude squared of the Fourrier Transform but since the frequency intervals are disjoint the squaring doesn't create any type of interaction between the two original spectral. So it's simply
$$P_{sum}(f) = \begin{cases}
 &amp; p_1\text { if } f_{11} &lt; f &lt; f_{12} \\ 
 &amp; p_2 \text{ if }  f_{21} &lt; f &lt; f_{22} \\
 &amp; 0 \text{ otherwise }
\end{cases} $$