Topological entropy of random walks on mapping class groups

For any pseudo-Anosov homeomorphism on a closed orientable surface S of genus greater than one, it is known by the work of Bers and Thurston that the topological entropy agrees with the translation distance on the Teichmüller space with respect to the Teichmüller metric. In this article, we consider random walks on the mapping class group of S. The drift of a random walk is defined as the translation distance of the random walk. We define the topological entropy of a random walk and prove that it almost surely agrees with the drift on the Teichmüller space with respect to the Teichmüller metric.