= Solution
For $t_1<t_2<t_3$, the condition from part a becomes
$$
(t_3-t_1)^\eta
=(t_2-t_1)^\eta+(t_3-t_2)^\eta.
$$
It holds for arbitrary positive time gaps exactly when $\eta=1$. The resulting <exponential covariance function> is the covariance of a stationary <Ornstein-Uhlenbeck process>, hence has the <Markov property>.
Writing $a_{32}=e^{-(t_3-t_2)/\tau}$, the predictive law is
$$
y_3\mid y_2,y_1
\sim N\!\left(\mu+a_{32}(y_2-\mu),,1-a_{32}^2\right).
$$
As $t_3\to\infty$, $a_{32}\to0$, so the predictive mean tends to the stationary mean $\mu$ and the predictive variance tends to the stationary variance $1$.
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