Symmetric random walks, including those with step sizes drawn from a Gaussian process with mean zero, have expectation 0 at any time t. Since there's no drift term, either walk won't be expected to slowly deviate from zero.
But the variance of your probability distribution of where you'll be at time t is linear in t. So say that your variance is v(t)=t. Then at t=1, there is a 32% chance that you'll be outside of the range (-1,1). As you can see, as t increases, you expected to drift further than further.
So while the expectation of x(t) may be 0 for all time, the expectation of |x(t)| scales like sqrt(t) (the standard deviation of the distribution).