Tracial smooth functions of non-commuting variables and the free Wasserstein manifold
Volume 580 / 2022
Abstract
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 $\mathscr{W}(\mathbb{R}^{*d})$ 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 $\mathscr{D}(\mathbb{R}^{*d})$ acts on $\mathscr{W}(\mathbb{R}^{*d})$ by transport, and the basic relationship between tangent vectors for $\mathscr{D}(\mathbb{R}^{*d})$ 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 Guionnet and Shlyakhtenko (2014), Dabrowski et al. (2021) and Jekel (2022), we prove the existence of smooth transport along any path $t \mapsto V_t$ when $V_t$ is sufficiently close to $(1/2) \sum_j {\rm tr}(x_j^2)$, 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 $\mathscr{W}(\mathbb{R}^{*d})$ and hence of the free heat equation, optimal transport equations, incompressible Euler equation, and inviscid Burgers’ equation.
