Free Stein Kernel and Moments maps

In this paper, we propose a free analogue to Fathi's construction of Stein kernels using moment maps (2019). This is possible for a class of measures called free moment measures that was introduced in the free case by Bahr and Boschert (2021), and by using the notion of free moment maps which are convex functions, solutions of a variant of the free Monge-Amp\`ere equation discovered by Guionnet and Shlyakhtenko (2012). We then show how regularity estimates in some weighted non-commutative Sobolev spaces on these maps control the transport distances to the semicircular law. We also prove in the one dimensional case a free analogue of the moment map version of the Cafarelli contraction theorem (2001), discovered by Klartag in the classical case (2014), and which leads to a uniform bound on the free moment Stein kernel. Finally, we discuss the applications of these results: we prove a stability result characterizing the semicircular distribution among a certain subclass of free Gibbs measures, the probabilistic interpretation of this free moment Stein kernel in terms of free diffusion processes, its connections with the theory of non-commutative Dirichlet forms, and a possible notion of a non-commutative (free) Hessian manifold associated with a free Gibbs measure, conjecturally having free analogue properties of a classical Hessian manifold associated with a log-concave measure and considered by Kolesnikov (2012), which has the striking property of having a Ricci curvature bounded from below by $1/2$.