The Poisson Boundary of Teichmüller Space

For a Markov operator on Teichmüller space commuting with the action of the mapping class group we prove convergence of sample paths of the associated Markov chain in the Thurston compactification and show that the Poisson boundary of the Markov operator can be identified with the space of projective measured foliations. The approach consists in using the authorsʹ results on the Poisson boundary of the mapping class group in combination with a discretization procedure based on a Harnack inequality for Markov operators on Teichmüller space.