Factors, Roots and Embeddings of Measures on Lie Groups

Potential Analysis, 2015

We study (weakly) continuous convolution semigroups of probability measures on a Lie group G or a homogeneous space G/K, where K is a compact subgroup. We show that such a convolution semigroup is the convolution product of its initial measure and a continuous convolution semigroup with initial measure at the identity of G or the origin of G/K. We will also obtain an extension of Dani-McCrudden's result on embedding an infinitely divisible probability measure in a continuous convolution semigroup on a Lie group to a homogeneous space.

On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure, these factors become independent random variables with explicit distributions. Beyond the known results on the orthogonal and unitary groups (O(n) and U (n)), we treat the symplectic case. In U (n), this induces a family of probability changes analogous to the biassing in the Ewens sampling formula known for the symmetric group. Then we study the spectral properties of these measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The associated orthogonal polynomials give rise, as n tends to infinity to a limit kernel at the singularity.

arXiv preprint arXiv:1304.7182, 2013

In this work we are going to study the dynamics of the linear automorphisms of a measure convolution algebra over a finite group, T (µ) = ν * µ. In order to understand an classify the asymptotic behavior of this dynamical system we provide an alternative to classical results, a very direct way to understand convergence of the sequence {ν n } n∈N , where G is a finite group, ν ∈ P(G) and ν n = ν * ... * ν n , trough the subgroup generated by his support.

Illinois Journal of Mathematics, 1994

A theorem of Siebert asserts that if µn(t) are semigroups of probability measures on a Lie group G, and Pn are the corresponding generating functionals, then µn(t), f − → n µ 0 (t), f , f ∈ C b (G), t > 0, implies π Pn u, v − → n π P 0 u, v , u ∈ C ∞ (π), v ∈ X, for every unitary representation π of G on a Hilbert space X, where C ∞ (π, X) denotes the space of smooth vectors for π. The aim of this note is to give a simple proof of the theorem and propose some improvements. In particular, we completely avoid employing unitary representations by showing simply that under the same hypothesis Pn, f − → n P 0 , f , f ∈ C 2 b (G). As a corollary, the above thesis of Siebert is extended to strongly continuous representations of G on Banach spaces.

2011

We describe certain sufficient conditions for an infinitely divisible probability measure on a class of connected Lie groups to be embeddable in a continuous one-parameter convolution semigroup of probability measures. (Theorem 1.3). This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain Lie groups, including the so called Walnut group (Corollary 1.5). The embeddability is concluded also under certain other conditions (Corollary 1.4 and Theorem 1.6).

2016

A theorem of Siebert asserts that if µn(t) are semigroups of probability measures on a Lie group G, and Pn are the corresponding generating functionals, then µn(t), f − → n µ 0 (t), f , f ∈ C b (G), t > 0, implies π Pn u, v − → n π P 0 u, v , u ∈ C ∞ (π), v ∈ X, for every unitary representation π of G on a Hilbert space X, where C ∞ (π, X) denotes the space of smooth vectors for π. The aim of this note is to give a simple proof of the theorem and propose some improvements. In particular, we completely avoid employing unitary representations by showing simply that under the same hypothesis Pn, f − → n P 0 , f , f ∈ C 2 b (G). As a corollary, the above thesis of Siebert is extended to strongly continuous representations of G on Banach spaces.

Mathematical Notes, 2018

Physica A: Statistical Mechanics and its Applications, 1990

The Markov jump processes on compact groups considered here are assumed to be invariant in the sense that the Markov transition probability function P,,, can be defined by a convolution semi-group (T,),," of probability measures on G as P,,, = e, * T,, where E, is the Dirac measure concentrated on X. This semi-group is shown to be a {e}-Poisson semi-group and is determined by its generating functional 4, a measure on G such that-d, is a Poisson form. The processes are ergodic and the stationary probability distribution is related to the Haar measure on a closed subgroup H of G. Several properties of these processes, especially the spectral analysis of the Markov semi-group of bounded linear operators in V(G), are derived by the use of Fourier analysis. It is also shown that the jump processes can approximate the invariant Feller processes on compact groups. A number of examples is provided for the discussion of further features and possible classification schemes.

Transactions of the American Mathematical Society

If G is a compact group with w(G) = a ≥ ω, we show the following results: (i) There exist direct products ξ<a G ξ , ξ<a H ξ of compact metric groups and continuous open surjections ξ<a G ξ p → G q → ξ<a H ξ with respect to Haar measure; and (ii) the Haar measure on G is Baire and at the same time Jordan isomorphic to the Haar measure on a direct product of compact Lie groups. Applications of the above results in measure theory are given.