If I understand correctly, white noise refers to a Brownian motion that is independent at different locations: $E[\dot{B}_x\dot{B}_{x'}] = \delta_{x-x'}$. Space-time white noise is independent in both time and space: $$ E[\dot{B}(x,t) \dot{B}(x',t')] = \delta_{x-x'} \delta_{t-t'} \,.$$ (is that right?)
Colored noise refers to noise that has some correlation with itself at different points and/or time. What exactly should that correlation be, and how can we define it?
I know that white noise/brownian motion can be expanded into Fourier series, i.e. something like $$ \frac{dW(t)}{dt} = \sum_{k=1}^n \eta_k \psi_k(t)$$ where $\eta_k \sim N(0,1)$ are iid and $\psi_k(t)$ are the Fourier basis functions. From this, how is colored noise defined? We could e.g. truncate the series, or introduce a weight term that makes makes larger-$k$ terms decay faster. What are the principles one would use in choosing different methods of approximation? I'm not sure how auto-correlation is related to the method of approximation.