Nonlinear Markov operators associated with symmetric Markov kernels and energy minimizing maps between singular spaces
The author considers harmonic maps defined on a space with measure \(m\) and a symmetric Markov kernel \(p\) on it, with values in a complete geodesic space of non-positive curvature in the sense of A. D. Alexandrov. Thus, the theme of the paper belongs to the theory of nonlinear Dirichlet forms and energy minimizing maps between singular spaces as developed, e.g. by \textit{J. Jost} [Calc. Var. Partial Differ. Equ. 2, 173-204 (1994; Zbl 0798.58021)] and \textit{N. Korevaar} and \textit{R. Schoen} [Commun. Anal. Geom. 1, 561-659 (1993; Zbl 0862.58004)]. It is proved in fact that the harmonic maps are exactly the minimizers of the energy. The paper specializes (to the case where \(p\) is symmetric with respect to the measure \(m\)) results obtained previously by the author: Nonlinear Markov operators, discrete heat flow, and harmonic maps between singular spaces, Potential Anal. 16, No. 4, 305-340 (2002). As such, it may be considered as a nonlinear generalization of the theory of symmetric Markov kernels; various properties of the linear heat flow carry over to the nonlinear case.
scientific article; zbMATH DE number 1655599