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The problem of the harmonic oscillator with a random small perturbation is known. We can write the equation as: $$\ddot{x}(t)+\eta^2x(t)=0$$
where $\eta^2=\omega^2+\epsilon{W_t}$ with $\epsilon>0$ and $W_t$: white gaussian noise. My question is: if the noise has a Rayleigh $pdf$:$$f(z;\sigma)=\frac{z}{\sigma^2}\exp(\frac{-z^2}{2\sigma^2})$$ with $z\ge0$, is it possible to find a solution for the $x(t)$?

Thanks in advance.

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