I have been wondering for a while now, and it my be a bad question, if a realization of a Gaussian process can be written and generated similar to an Ito diffusion.
From what I know, a Gaussian processes is generated from the covariance kernel K(x,x') and so can Brownian motion.
This is the Ito form: $ dX(t)=\mu dt+\sigma dW(t) $
So because diffusion has time, then my guess the gaussian kernel would be K(x,t)