Summary: Using new spaces of tracial non-commutative smooth functions, we formulate a free probabilistic analog of the Wasserstein manifold on \(\mathbb{R}^d\) (the formal Riemannian manifold of smooth probability densities on \(\mathbb{R}^d)\), and we use it to study smooth non-commutative transport of measure. The points of the free Wasserstein manifold \(\mathscr{W}(\mathbb{R}^{\ast d})\) are smooth tracial non-commutative functions \(V\) with quadratic growth at \(\infty \), which correspond to minus the log-density in the classical setting. The space of non-commutative diffeomorphisms \(\mathscr{D}(\mathbb{R}^{\ast d})\) acts on \(\mathscr{W}(\mathbb{R}^{\ast d})\) by transport, and the basic relationship between tangent vectors for \(\mathscr{D}(\mathbb{R}^{\ast d})\) and tangent vectors for \(\mathscr{W}(\mathbb{R}^{*d})\) is described using the Laplacian \(L_V\) associated to \(V\) and its pseudo-inverse \(\Psi_V\) (when defined).
Following similar arguments to those of A. Guionnet and D. Shlyakhtenko [Invent. Math. 197, No. 3, 613–661 (2014; Zbl 1312.46059)], Y. Dabrowski et al. [N. Z. J. Math. 52, 259–359 (2021; Zbl 1484.46069)] and D. Jekel [Int. Math. Res. Not. 2022, No 6, 4514–4619 (2022; Zbl 1496.46067)], we prove the existence of smooth transport along any path \(t \mapsto V_t\) when \(V_t\) is sufficiently close to \((1/2) \sum_j\mathrm{tr}(x_j^2)\), as well as smooth triangular transport. The two main ingredients are (1) the construction of \(\Psi_V\) through the heat semigroup and (2) the theory of free Gibbs laws, that is, non-commutative laws maximizing the free entropy minus the expectation with respect to \(V\). We conclude with a mostly heuristic discussion of the smooth structure on \(\mathscr{W}(\mathbb{R}^{\ast d})\) and hence of the free heat equation, optimal transport equations, incompressible Euler equation, and inviscid Burgers’ equation.