Nonlinear Markov Operators, Discrete Heat Flow, and Harmonic Maps Between Singular Spaces

Abstract

We develop a theory of harmonic maps f:M→N between singular spaces M and N. The target will be a complete metric space (N,d) of nonpositive curvature in the sense of A. D. Alexandrov. The domain will be a measurable space (M,ℳ) with a given Markov kernel p(x,dy) on it. Given a measurable map f:M→N, we define a new map Pf:M→N in the following way: for each x∈M, the point Pf(x)∈N is the barycenter of the probability measure p(x,f −1(dy)) on N. The map f is called harmonic on D⊂M if Pf=f on D. Our theory is a nonlinear generalization of the theory of Markov kernels and Markov chains on M. It allows to construct harmonic maps by an explicit nonlinear Markov chain algorithm (which under suitable conditions converges exponentially fast). Many smoothing and contraction properties of the linear Markov operator P M,R carry over to the nonlinear Markov operator P=P M,N . For instance, if the underlying Markov kernel has the strong Lipschitz Feller property then all harmonic maps will be Lipschitz continuous.