I might be confused about something. 

Consider doing inference on $Y'\mid X',Y,X$ using standard Gaussian Process Regression with 1d $Y$ and 1d $X$. Suppose $X$ is time-like (target is stationary or shift invariant and that the covariance grows like $\Delta X$. This would imply a good kernel is something like $k(x, x') \propto \frac{e^{- \frac{\alpha}{|x - x'|}}}{| x - x'|}$ yet we often see the squared exponential used as a default for stationary GPR kernels which is $e^{- \frac{(x - x')^2}{2 \sigma^2}}$. 

Obviously the kernel doesn't make much sense because it tends to zero $x$ goes to $x'$. What is wrong with the reasoning? Why does "time like" mean using squared exponential?

Another way of asking this question is what kernel corresponds to the GPR being a brownian bridge density?