In page 7 from the notes "An introduction to SPDE's" from Walsh, one reads:
If we compute the covariance of $V_s$ we obtain (for $s\leq t$)
$$ Cov (V_s, V_t) = Cov(U_{s,a +bs}, U_{t,a +bt}) = e^{-|s - t|} e^{-|b||s-t|} = e^{-(1+|b|)|s-t|} \tag{*}$$
Compare with wikipedia's value for the covariance of Ornstein-Uhlenbeck processes covariance function for Brownian motion
Also checking Covariance of Ornstein - Uhlenbeck Process and Covariance of Ornstein-Uhlenbeck process, the values expressed there do not seem to match my calculations.
Are the computation in $(*)$ correct? This characterizes the process as an Ornstein-Uhlenbeck process? Why do we see these different expressions for the covariances?