一个经典与自由独立概念的连续半群

Traffic distributions and independence: permutation invariant random matrices and the three notions of independence

Voiculescu's notion of asymptotic free independence is known for a large class of random matrices including independent unitary invariant matrices. This notion is extended for independent random matrices invariant in law by conjugation by permutation matrices. This fact leads naturally to an extension of free probability, formalized under the notions of traffic probability. We first establish this construction for random matrices. We define the traffic distribution of random matrices, which is richer than the *-distribution of free probability. The knowledge of the individual traffic distribut…

On a notion of independence proposed by Teddy Seidenfeld

Teddy Seidenfeld has been arguing for quite a long time that binary preference models are not powerful enough to deal with a number of crucial aspects of imprecision and indeterminacy in uncertain inference and decision making. It is at his insistence that we initiated our study of so-called sets of desirable option sets, which we have argued elsewhere provides an elegant and powerful approach to dealing with general, binary as well as non-binary, decision-making under uncertainty. We use this approach here to explore an interesting notion of irrelevance (and independence), first suggested by …

Strict continuity of the transition semigroup for the solution of a well-posed martingale problem

In this note we connect the notion of solutions of a martingale problem to the notion of a strongly continuous and locally equi-continuous semigroup on the space of bounded continuous functions equipped with the strict topology. This extends the classical connection of semigroups to Markov processes that was used successfully in the context of compact spaces to the context of Polish spaces. In addition, we consider the context of locally compact spaces and show how the transition semigroup on the space of functions vanishing at infinity can be extended to the space of bounded continuous functi…

循环单调独立与单调独立的矩阵模型

Matrix models for cyclic monotone and monotone independences

本文证明同一随机矩阵模型在适当状态下同时作为循环单调独立和单调独立的模型，从而为两种独立概念提供统一框架。

Categorial Independence and Lévy Processes

We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial Lévy processes on every tensor category with initial unit object and present a construction generalizing the reconstruction of a Lévy process from its …

量子Levy过程理论

The Theory of Quantum Levy Processes

介绍了量子Lévy过程的各种最新结果。第一部分介绍了对合双代数上的Lévy过程理论。用于这些过程的独立性概念是张量独立性，它概括了古典概率中使用的独立性概念，并对应于量子物理学中的独立观测量。在量子概率中，还存在其他独立性的概念，并且Lévy过程也可以定义为五个所谓的普遍独立性。这是第二部分的主题。特别是，它表明，布尔，单调和反单调的独立性可以减少到张量的独立性。最后，在第三部分中，考虑了几类特别感兴趣的量子Lévy过程，例如，实李代数上的Lévy过程或辫状空间上的布朗运动。这些过程的几个应用也被提出。

$\mathbb{Z}_2$分次$C^*$代数的$C^*$独立性

$C^*$-independence for $\mathbb{Z}_2$-graded $C^*$-algebras

本文研究$\mathbb{Z}_2$分次$C^*$代数的$C^*$独立性，建立分次von Neumann代数上多种统计独立性概念的关系，并刻画分次核性质。

Pressures for multi-potentials in semigroup dynamics

We study several notions of topological pressure and capacities for multi-potentials $Φ\in \mathcal C(X;\mathbb R)^m$, with respect to finitely generated continuous semigroups $G$ on a compact metric space $X$. We introduce the amalgamated pressure, the condensed pressure, the trajectory pressure, the exhaustive pressure, and the respective capacities on non-compact sets $Y$, for multi-potentials $Φ$. This is done by using Carathéodory-Pesin structures. Several properties of these types of pressure, and relations between them are explored. The inverse limit of the semigroup and its relations t…

BMT 独立性

BMT Independence

本文提出 BMT 独立性以统一并推广 boolean、monotone、tensor 及 BM 独立性，利用有向图确定混合矩计算，并建立相应极限定理与 Hilbert 空间有界算子构造。

Conditionally Bi-Free Independence for Pairs of Algebras

In this paper, the notion of conditionally bi-free independence for pairs of algebras is introduced. The notion of conditional $(\ell, r)$-cumulants are introduced and it is demonstrated that conditionally bi-free independence is equivalent to mixed cumulants. Furthermore, limit theorems for the additive conditionally bi-free convolution are studied using both combinatorial and analytic techniques. In particular, a conditionally bi-free partial $\mathcal{R}$-transform is constructed and a conditionally bi-free analogue of the Lévy-Hinčin formula for planar Borel probability measures is derived…

Bi-Boolean independence for pairs of algebras

In this paper, the notion of bi-Boolean independence for non-unital pairs of algebras is introduced thereby extending the notion of Boolean independence to pairs of algebras. The notion of B-$(\ell, r)$-cumulants is defined via a bi-Boolean moment-cumulant formula over the lattice of bi-interval partitions, and it is demonstrated that bi-Boolean independence is equivalent to the vanishing of mixed B-$(\ell, r)$-cumulants. Furthermore, some of the simplest bi-Boolean convolutions are considered, and a bi-Boolean partial $η$-transform is constructed for the study of limit theorems and infinite d…

什么是随机独立性？

What is Stochastic Independence?

定义了带投影或带包含的张量积的概念。它表明，随机独立性的定义依赖于这样的结构，独立性可以定义在任意类别的张量积包含或投影。在这种情况下，Muraki、Ben Ghorbal和Schürmann对量子随机独立性的分类就变成了对代数概率空间和非单位代数概率空间的张量积的分类。还介绍了减少一个独立到另一个的概念。作为例子，费米独立性和布尔，单调，反单调的独立性张量独立性的减少。

Generalized Schrödinger semigroups on infinite graphs

With appropriate notions of Hermitian vector bundles and connections over weighted graphs which we allow to be locally infinite, we prove Feynman-Kac-type representations for the corresponding semigroups and derive several applications thereof.

关于Forré的条件独立性概念及连续变量的因果演算的注记

Notes on Forré's Notion of Conditional Independence and Causal Calculus for Continuous Variables

本文探讨Forré提出的过渡条件独立性概念，扩展了Richardson等人在一般测度理论下的ID算法形式化，强调其与文献的联系及因果演算的细微之处。

Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution $μ$ on $k$-sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $S$, the relative entropy of a single element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mix…