Nonlinear Markov operators associated with symmetric Markov kernels and energy minimizing maps between singular spaces

Abstract

We present a theory of harmonic maps $f:M\to N$ where the target is a complete geodesic space (N,d) of nonpositive curvature in the sense of A. D. Alexandrov and the domain is a measure space $(M,{\cal M},m)$ with a symmetric Markov kernel p on it. Our theory is a nonlinear generalization of the theory of symmetric Markov kernels and reversible Markov chains on M. It can also be regarded as a particular case of the theory of generalized (= nonlinear) Dirichlet forms and energy minimizing maps between singular spaces, initiated by Jost (1994) and Korevaar, Schoen (1993) and developed further by Jost (1997a), (1998) and Sturm (1997). We investigate the discrete and continuous time heat flow generated by p and show that various properties of the linear heat flow carry over to this nonlinear heat flow. In particular, we study harmonic maps, i.e. maps which are invariant under the heat flow. These maps are identified with the minimizers of the energy.