On continuity of drifts of the mapping class group

A random walk on a countable group $G$ acting on a metric space $X$ gives a characteristic called the drift which depends only on the transition probability measure $\mu$ of the random walk. The drift is the `translation distance' of the random walk. In this paper, we prove that the drift varies continuously with the transition probability measures, under the assumption that the distance and the horofunctions on $X$ are expressed by certain ratios. As an example, we consider the mapping class group $\mathrm{MCG}(S)$ acting on the Teichm\"uller space. By using north-south dynamics, we also consider the continuity of the drift for a sequence converging to a Dirac measure. As an appendix, we prove that the asymptotic entropy of the random walks on $\mathrm{MCG}(S)$ varies continuously.