*Preprint*

**Inserted:** 25 oct 2017

**Last Updated:** 25 oct 2017

**Year:** 2017

**Abstract:**

We propose a definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings $\mathbf{\mu} : \Omega \to (\mathcal{P}(D), W_2)$ defined over a subset $\Omega$ of $\mathbf{R}^p$ and valued in the space $\mathcal{P}(D)$ of probability measures on a compact convex subset $D$ of $\mathbf{R}^q$ 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. We study harmonic mappings, i.e. minimizers of the Dirichlet energy provided that the values on the boundary $\partial \Omega$ are fixed. The notion of constant-speed geodesics in the Wasserstein space is recovered by taking for $\Omega$ a segment of $\mathbf{R}$. As the Wasserstein space $(\mathcal{P}(D), W_2)$ is positively curved in the sense of Alexandrov we cannot apply the theory of Koorevaar, Schoen and Jost and we use instead optimal transport based arguments. We manage to get existence of a harmonic mapping provided that the boundary values are Lipschitz on $\partial \Omega$, uniqueness is an open question. If $\Omega$ is a segment of $\mathbf{R}$, it is known that a curve valued in the Wasserstein space 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 $\Omega \to \mathcal{P}(D)$ cannot be represented as the superposition of mappings $\Omega \to D$. We are able to show a Ishihara-type property: the composition $F \circ \mathbf{\mu}$ of a function $F : \mathcal{P}(D) \to \mathbf{R}$ convex along generalized geodesics and an harmonic mapping $\mathbf{\mu} : \Omega \to \mathcal{P}(D)$ is a subharmonic real-valued function. We also study the special case where we restrict ourselves to a given location-scatter family (a finite-dimensional and geodesically convex submanifold of $(\mathcal{P}(D), W_2)$ 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.

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