Title: Free Stein kernels, free moment maps, and higher order derivatives Show Chinese title

Title: 自由Stein核，自由矩映射和高阶导数

Abstract: In this work, we describe new constructions of free Stein kernels. Firstly, in dimension one, we propose a free analog to the construction of Stein kernels using moment maps as the one proposed by Fathi. This will be possible for a class of measures called the free moment measures via the notion of free moment map (convex functions), introduced in the free case by Bahr and Boschert. In a second time, we introduce the notion of higher-order free Stein kernels relative to a potential, which can be thought as the free counterpart of a recent and powerful idea introduced in the classical case by Fathi, and which generalize the notion of free Stein kernels by introducing higher-order derivatives of test functions (in our context noncommutative polynomials). We then focus our attention to the case of homothetic semicircular potentials. We prove as in the classical case, that their existence implies moments constraints. Finally, we relate these discrepancies to various metrics: the free (quadratic) Wasserstein distance, the relative free Fisher information along the Ornstein-Uhlenbeck flow or the relative non-microstates free entropy. Finally, as an important application, we provide new rates of convergences in the entropic free CLT under higher moments constraints. Show Chinese abstract

Abstract: 在本工作中，我们描述了自由Stein核的新构造。首先，在一维情况下，我们提出了一种使用矩映射的Stein核构造的自由类似物，正如Fathi所提出的那样。这将通过称为自由矩测度的一类测度实现，这是通过自由矩映射（凸函数）的概念实现的，该概念由Bahr和Boschert在自由情况下引入。其次，我们引入了相对于势能的高阶自由Stein核的概念，这可以看作是Fathi在经典情况下引入的一个最近且强大的想法的自由对应物，并通过引入测试函数的高阶导数（在我们的上下文中为非交换多项式）来推广自由Stein核的概念。随后，我们将注意力集中在相似半圆势的情况上。我们证明了与经典情况类似，它们的存在性意味着矩约束。最后，我们将这些差异与各种度量相关联：自由（二次）Wasserstein距离、Ornstein-Uhlenbeck流上的相对自由Fisher信息或相对非微观状态自由熵。最后，作为重要的应用，我们在更高矩约束下的熵自由中心极限定理中提供了新的收敛速率。