autocorrelation - How do we compute distrubtions of the value of a random process conditional on initial conditions?

Suppose I have a stationary process $\phi(t)$ with a known autocorrelation function

$$A(\tau) \equiv \langle \phi(0) \phi(\tau) \rangle$$

and suppose I also know that $\phi(t)$ is Gaussian distributed.

If I have a particular realization of the process in which $\phi(0)=\phi_0$, what is the conditional distribution of $\phi(t)$ for later times in that realization?