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## Dissertationes Mathematicae

Artykuły w formacie PDF dostępne są dla subskrybentów, którzy zapłacili za dostęp online, po podpisaniu licencji Licencja użytkownika instytucjonalnego. Czasopisma do 2009 są ogólnodostępne (bezpłatnie).

## Tracial smooth functions of non-commuting variables and the free Wasserstein manifold

### Tom 580 / 2022

Dissertationes Mathematicae 580 (2022), 1-150 MSC: Primary 46L54; Secondary 46L52, 35Q49, 94A17, 58D99. DOI: 10.4064/dm843-10-2021 Opublikowany online: 2 June 2022

#### Streszczenie

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.

#### Autorzy

• David JekelDepartment of Mathematics
University of California, San Diego 9500 Gilman Drive # 0112
La Jolla, CA 92093-0112, USA
e-mail
• Wuchen LiDepartment of Mathematics
University of South Carolina
1523 Greene Street
Columbia, SC 29208-4014, USA
e-mail
• Dimitri ShlyakhtenkoDepartment of Mathematics
University of California, Los Angeles
520 Portola Plaza
Box 951555
Los Angeles, CA 90095-1555, USA
e-mail

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