This question is influenced by this paper.
Consider a random field $Z(t,x)$ with a correlation structure given by
$$d\langle Z(t,x),Z(t,y)\rangle =c(x,y)\,dt.$$
For example, a choice for the correlation function can be $c(x,y)=ae^{-k|x-y|}.$
My question is, what is the meaning of the correlation structure $d\langle Z(t,x),Z(t,y)\rangle=c(x,y)\,dt$? Is $c(x,y)$ an instantaneous correlation between the random variables $Z(t,x)$ and $Z(t,y)$?. How (or why) is the cross-variation $\langle Z(t,x),Z(t,y)\rangle$ related to the correlation structure? Any help is appreciated!