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Stochastic Aspects of Nonlinear Refinement Schemes

This article develops a stochastic viewpoint on nonlinear refinement algorithms designed to construct continuous objects from discrete samples in a global NPC (nonpositive curvature) space. A particular focus lies on convergence properties of so-called barycentric subdivision schemes, a class of refinement algorithms acting on input data via weighted averages. An essential observation is the characterization of these kinds of schemes as nonlinear Markov semigroups. Exploiting this feature, it is proven that convergence on arbitrary NPC spaces is equivalent to convergence for real-valued input data. Furthermore, a strong law of large numbers leads to certain structure-preserving properties for barycentric schemes on the space of positive definite matrices. A concluding section addresses the relationship between the convergence properties of a scheme and its so-called characteristic Markov chain.