The manifold-valued Dirichlet problem for symmetric diffusions (Q1841543)

The construction of harmonic maps between Riemannian manifolds \(M\) and \(N\) as energy minimizing maps is extended for the Dirichlet problem to the case where the Riemannian energy functional on \(M\) is replaced by a more general Dirichlet form. The main Dirichlet problem is proved in two steps, by solving the variational problem and by probabilistic analysis involving a Dirichlet process. Weakly harmonic maps that send the diffusion to \(N\)-valued martingales are obtained.