  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 $G$.
