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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!

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    The same method should work. It's a little tricky to find the points which are distance $\epsilon$ away from a given point in the disk model. One way is to translate a circle from the center.2011-04-05
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    Yes, that's correct. And it is transient, but how can you tell this from running a simulation for finite time? Maybe you need to run it longer?2011-04-05
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    I do not have the reputation to comment but shouldn't be the steps be Gaussian random variables of zero mean and variance $\epsilon$ instead of steps of constant length? Does it makes a difference?2011-04-05
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    It makes no difference in the limit as the step size goes to zero (and the number of steps goes to infinity).2011-04-06
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    @user9126: please do not use answers to comment.2011-04-06
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    This can be generalized quite a bit. See, for instance, Ming Liao, _Lévy Processes in Lie Groups_, Cambridge University Press, Cambridge Tracts in Mathematics, 162. Hyperbolic spaces have nice presentations as $SO^+(1,n)/SO(n)$ so can be handled with essentially the same machinery.2011-04-19

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