自由Stein核与矩映射
Free Stein Kernel and Moments maps
AI总结：
本文利用自由矩映射构造了自由Stein核，建立其正则性估计与到半圆律的传输距离之间的联系，证明一维自由矩映射版Cafarelli收缩定理，并应用于自由Gibbs测度的稳定性、自由扩散过程及非交换Hessian流形研究。
摘要：
在本文中，我们利用矩映射提出了Fathi（2019）Stein核构造的一个自由类似物。这对于一类被称为自由矩测度的测度是可能的，该测度在自由情形下由Bahr和Boschert（2021）引入，并且利用了自由矩映射的概念，这些自由矩映射是凸函数，是Guionnet和Shlyakhtenko（2012）发现的自由Monge-Ampère方程的一个变体的解。然后我们展示了这些映射在某些加权非交换Sobolev空间中的正则性估计如何控制到半圆律的传输距离。我们还在维数为一的情形下证明了Cafarelli收缩定理（2001）的矩映射版本的一个自由类似物，该版本由Klartag在经典情形下发现（2014），并且它导致了对自由矩Stein核的一致有界性。最后，我们讨论这些结果的应用：我们证明了一个稳定性结果，刻画了自由Gibbs测度的某个子类中的半圆分布，给出了该自由矩Stein核在自由扩散过程方面的概率解释，它与非交换Dirichlet型理论的联系，以及与自由Gibbs测度相关联的非交换（自由）Hessian流形的一个可能概念，该流形被猜想具有与对数凹测度相关联并由Kolesnikov（2012）考虑的经典Hessian流形的自由类似性质，该经典流形具有Ricci曲率下界为$1/2$的显著性质。
英文摘要：
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-Ampère 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 $1/2$.
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