Let's take an input-output stochastic system with a given noise. First of all we should define the noise process so we will have:
$$d x_t=-\theta x_t d t+\sigma d W_t$$ where the $d W_t$ is a standard wiener process(Brownian motion).
Multiply $e^{\theta t}$ to both sides so we will have:
$$e^{\theta t}*d x_t+e^{\theta t} \theta x_t d t = e^{\theta t}* \sigma d W_t$$ $$d(x_t e^{\theta t}) = e^{\theta t}\sigma dW_t$$ Without loss of generality take $X_0 = N(0,\sigma '^2)$, so then:
$$x_t e^{\theta t} =x_0 + \sigma \int e^{\theta s} dW_s$$ $$x_t(u) =X_0 e^{-\beta t}+ \sigma \int_{0}^{u} e^{\theta (s-t)} dW_s$$
Now we define the noise as below: $$N_t=\xi(t)=\int_0^t x(s) d s$$
How can I write $N_t$?