I have a linear dynamical system, whose output $o(t)$ in response to an input $i(t)$ is given by
$$ o(t) = \int_{-\infty}^tK(t-u)i(u)du, $$
where $K(t)$ is a response function, $ K(w) = e^{-w}(w-w/2)$, where $ w = t/\gamma $, and $\gamma$ is an internal time scale.
I would like to know what is the timescale of fluctuations of $o(t)$ is, when $i(t)$ is an Ornstein-Uhlenbeck process with correlation length $\mu$, i.e. $\langle i(t)i(t-t')\rangle = \exp(-t'/\mu)$.
I am quite new to this field, could someone help how to do this?
I thought about two possible ways:
1) If the input was deterministic, I would take the Fourier transform $$ \tilde{o}(\omega) = \tilde{K}(\omega)\tilde{i}(\omega). $$ Then I can easily compute $\tilde{K}$ in my case. However, what is the FT of the OU stochastic process, $\tilde{i}$?
2) I also considered, writing $o(t)$ and $i(t)$ as a linear dynamical system $$ \dot{o}(t) = Ao(t) + Bi(t)\\\dot{i}(t) = -i(t)/\mu + \eta(t)/\mu, $$ where $ \eta$ is white noise and A and B can be computed and the eigenvalues of $A$ are just $-1/\gamma$.