  This article settles the convergence question for multivariate barycentric
subdivision schemes with nonnegative masks on complete metric spaces of
nonpositive Alexandrov curvature, also known as Hadamard spaces. We establish a
link between these types of refinement algorithms and the theory of Markov
chains by characterizing barycentric subdivision schemes as nonlinear Markov
semigroups. Exploiting this connection, we subsequently prove that any such
scheme converges on arbitrary Hadamard spaces if and only if it converges for
real valued input data. Moreover, we generalize a characterization of
convergence from the linear theory, and consider approximation qualities of
barycentric subdivision schemes. A concluding section addresses the
relationship between the convergence properties of a scheme and its so-called
characteristic Markov chain.
