Nonlinear Markov semigroups and refinement schemes on metric spaces

Abstract: 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.