How to find a differential equation from a correlation function

Question

I'm trying to find a general method for deriving the differential equation of a noise field if given its auto-correlation function. For example, I know that if I have a noise field $$\langle \xi(t)\xi(s)\rangle = \frac{\epsilon^2}{2\tau}e^{-|t-s|/\tau}$$ then the differential equation for $\xi$ is given by $$\frac{d\xi}{dt} = \frac{-1}{\tau}\xi+\frac{\epsilon}{\tau}\eta$$ where $\eta$ is a white noise field; $\langle\eta(t)\eta(s)=\delta(t-s),\,\langle\eta(t)\rangle=0.$ This seems to be a common case, but I can't find any derivation of the differential equation. I'm working with more general noises. If I know that $$\langle w(t) w(s)\rangle = f(t,s)$$ how do I find $\frac{dw}{dt}$?