  In this article we prove existence of the asymptotic entropy for isotropic
random walks on regular Fuchsian buildings. Moreover, we give formulae for the
asymptotic entropy, and prove that it is equal to the rate of escape of the
random walk with respect to the Green distance. When the building arises from a
Fuchsian Kac-Moody group our results imply results for random walks induced by
bi-invariant measures on these groups, however our results are proven in the
general setting without the assumption of any group acting on the building. The
main idea is to consider the retraction of the isotropic random walk onto an
apartment of the building, to prove existence of the asymptotic entropy for
this retracted walk, and to `lift' this in order to deduce the existence of the
entropy for the random walk on the building.
