If I want to simulate Brownian motion in the Euclidean space I can simulate it by a point that is moving a distance $\epsilon$ in an arbitrary direction then it randomly choose a new direction and moves a distance $\epsilon$ again and so on. The smaller the $\epsilon$ the closer the simulation will be to the real Brownian motion.

How can I simulate Brownian motion in the hyperbolic space (Poincare Disk model for instance)? Does the same work here where I replace the Euclidean distance by the hyperbolic distance? My intuition is yes but when I did the simulation the random walk do not seem to be transient but it should be!