Contracting isometries and differentiability of the escape rate

Let G be a countable group whose action on a metric space X involves a contracting isometry. This setting naturally encompasses groups acting on Gromov hyperbolic spaces, Teichm\"uller space, Culler-Vogtmann Outer space and CAT (0) spaces. We discuss continuity and differentiability of the escape rate of random walks on G. For mapping class groups, relatively hyperbolic groups, CAT (-1) groups and CAT (0) cubical groups, we further discuss analyticity of the escape rate. Finally, assuming that the action of G on X is weakly properly discontinuous (WPD), we discuss continuity of the asymptotic entropy of random walks on G.