In this article, page 4 of the linked pdf file, Lalley and Sellke claim that a hyperbolic Brownian motion can be obtained by time-changing a 2-dimensional Euclidean Brownian motion, conditioned to stop at the $x$-axis. In their construction, it is not clear how the time-change is implemented.
Can anyone suggest references that explore this idea of time-change in detail? Particularly for the Poincaré disk model.