How do we compute distrubtions of the value of a random process conditional on initial conditions?
"I know that $\phi(t)$ is Gaussian distributed" is not the same as saying
"$\{\phi(t)\}$ is a Gaussian process" but I will assume that the latter is meant. With this assumption, and the additional assumption that the process is
wide-sense-stationary (the autocorrelation function is listed as having only one argument), the process is strictly stationary and thus
$\phi(t)$ and $\phi(0)$ are jointly Gaussian random variables. They have
the same mean $\mu$ and variance $\sigma^2$, and 
their correlation coefficient is $\rho$ where
$$\mu = \sqrt{\lim_{t\to \infty} A(t)}, \qquad \sigma^2 = A(0)-\mu^2,
\qquad \rho(t) = \frac{A(t)-\mu^2}{\sigma^2}.$$
Consequently the conditional distribution of $\phi(t)$ given
that $\phi(0)$ has taken on value $\phi_0$ is Gaussian
with mean $\mu + \rho(t)(\phi_0 - \mu) = \rho(t) \phi_0 + (1-\rho(t))\mu$
and variance $\sigma^2(1-\rho(t)^2)$. Note that as $t\to\infty$, 
the conditional distribution of $\phi(t)$ approaches the
unconditional distribution of $\phi(t)$: the distant past affects
the present state less and less.

For more information about Gaussian processes, see, for example,
this answer to a
different question.