Deformations of convolution semigroups on commutative hypergroups

Bessel convolutions on matrix cones: Algebraic properties and random walks

Bessel-type convolution algebras of bounded Borel measures on the matrix cones of positive semidefinite $q\times q$-matrices over $\mathbb R, \mathbb C, \mathbb H$ were introduced recently by Rösler. These convolutions depend on some continuous parameter, generate commutative hypergroup structures and have Bessel functions of matrix argument as characters. Here, we first study the rich algebraic structure of these hypergroups. In particular, the subhypergroups and automorphisms are classified, and we show that each quotient by a subhypergroup carries a hypergroup structure of the same type. Th…

Dunkl理论、卷积代数及相关马尔可夫过程

Dunkl theory, convolution algebras, and related Markov processes

本文介绍有理Dunkl算子及相关马尔可夫过程理论，强调与欧几里得对称空间和Bessel超群的联系，并讨论乘积公式、超群结构及以Dunkl拉普拉斯为生成元的扩散-反射过程。

The exponential map in non-commutative probability

The wrapping transformation $W$ is a homomorphism from the semigroup of probability measures on the real line, with the convolution operation, to the semigroup of probability measures on the circle, with the multiplicative convolution operation. We show that on a large class $\mathcal{L}$ of measures, $W$ also transforms the three non-commutative convolutions---free, Boolean, and monotone---to their multiplicative counterparts. Moreover, the restriction of $W$ to $\mathcal{L}$ preserves various qualitative properties of measures and triangular arrays. We use these facts to give short proofs of…

Operator-Valued Monotone Convolution Semigroups and an Extension of the Bercovici-Pata Bijection

In a 1999 paper, Bercovici and Pata showed that a natural bijection between the classically, free and Boolean infinitely divisible measures held at the level of limit theorems of triangular arrays. This result was extended to include monotone convolution by the authors. In recent years, operator-valued versions of free, Boolean and monotone probability have also been developed. Belinschi, Popa and Vinnikov showed that the Bercovici-Pata bijection holds for the operator-valued versions of free and Boolean probability. In this article, we extend the bijection to include monotone probability theo…

The Absolute of finitely generated groups: I. Commutative groups

We give a complete description of the absolute of commutative finitely generated groups and semigroups. The absolute (previously called the exit boundary) is a further elaboration of the notion of the boundary of a random walk on a group (the Poisson--Furstenberg boundary); namely, the absolute of a (semi)group is the set of ergodic central measures on the compactum of all infinite trajectories of a simple random walk on the group. Related notions have been discussed in the probability literature: Martin boundary, entrance and exit boundaries (Dynkin), central measures on path spaces of graphs…