text: We propose a definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings mu : Omega -> (P(D), W ) defined over a subset Omega of R^p and valued in the space P(D) of probability measures on a compact convex subset D of R^q endowed with the quadratic Wasserstein distance. Our definition relies on a straightforward generalization of the Benamou-Brenier formula (already introduced by Brenier) but is also equivalent to the definition of Koorevaar, Schoen and Jost as limit of approximate Dirichlet energies, and to the definition of Reshetnyak of Sobolev spaces valued in metric spaces. We study harmonic mappings, i.e. minimizers of the Dirichlet energy provided that the values on the boundary d Omega are fixed. The notion of constant-speed geodesics in the Wasserstein space is recovered by taking for Omega a segment of R. As the Wasserstein space (P(D), W ) is positively curved in the sense of Alexandrov we cannot apply the theory of Koorevaar, Schoen and Jost and we use instead arguments based on optimal transport. We manage to get existence of harmonic mappings provided that the boundary values are Lipschitz on d Omega, uniqueness is an open question. If Omega is a segment of R, it is known that a curve valued in the Wasserstein space P(D) can be seen as a superposition of curves valued in D. We show that it is no longer the case in higher dimensions: a generic mapping Omega -> P(D) cannot be represented as the superposition of mappings Omega -> D. We are able to show a Ishihara-type property: the composition F(mu) of a function F : P(D) -> R convex along generalized geodesics and a harmonic mapping mu : Omega -> P(D) is a subharmonic real-valued function. We also study the special case where we restrict ourselves to a given family of elliptically contoured distributions (a finite-dimensional and geodesically convex submanifold of (P(D), W ) which generalizes the case of Gaussian measures) and show that it boils down to harmonic mappings valued in the Riemannian manifold of symmetric matrices endowed with the distance coming from optimal transport.
  @cite_2: We develop a general mathematical framework for variational problems where the unknown function assumes values in the space of probability measures on some metric space. We study weak and strong topologies and define a total variation seminorm for functions taking values in a Banach space. The seminorm penalizes jumps and is rotationally invariant under certain conditions. We prove existence of a minimizer for a class of variational problems based on this formulation of total variation, and provide an example where uniqueness fails to hold. Employing the Kan -torovich-Rubinstein transport norm from the theory of optimal transport, we propose a variational approach for the restoration of orientation distribution function (ODF)-valued images, as commonly used in Diffusion MRI. We demonstrate that the approach is numerically feasible on several data sets.
summary: Finally, in @cite_2 the authors also study mappings valued in the space of probability measures, but are rather interested in the bounded variation norm (the integral of the norm of the gradient) than in the Dirichlet energy. Their provide applications to the denoising of measure-valued images.