As I understand, Brownian noise can be simulated by the process
$$x_{n+1}=x_n+R_n$$
where $R\sim U[-a,a]$. The expected value for $x_n$ is then $x_0$. But $\text{Var} x_n\to\infty$ as $n\to\infty$ which makes it unsuitable when $x_n$ for some reason must stay bounded. A work-around that seems to make the variance limited is to add a damping term:
$$x_{n+1}^\prime=x_n^\prime+R_n-\xi\left(x_n^\prime - x_0\right),\quad\xi\in[0, 1)$$
This makes $x_n^\prime\to x_0$ exponentially. How good is the alternative formula? What is the new expectation value for $x_n^\prime$. What is the variance?
"good" in this case is how far the different spectra are from each other
$$\left(\int_\omega\left|\sum_{k=-\infty}^\infty\left( x_k -x_k^\prime\right)\exp(-ik\omega) \right|^2 d\omega\right)^{-1}$$
One over because a small difference is good and a large difference is bad.