scholarly journals Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces

2011 ◽  
Vol 32 (4) ◽  
pp. 1313-1349 ◽  
Author(s):  
Y. GUIVARC’H ◽  
C. R. E. RAJA
Keyword(s):  
Random Walks ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Local Fields ◽  
Quadratic Growth ◽  
Linear Groups ◽  
Compact Groups ◽  
Large Classes ◽  
Skew Products ◽  
Recurrent Random Walk

AbstractWe discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular, we show that a closed subgroup of a product of finitely many linear groups over local fields supports an adapted recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces and for associated homeomorphisms with infinite invariant measure. The structural properties of closed subgroups of linear groups over local fields and the properties of group actions with respect to certain Radon measures associated with random walks play an important role in the proofs.

A Class of Abstract Linear Representations for Convolution Function Algebras over Homogeneous Spaces of Compact Groups

2018 ◽  
Vol 70 (1) ◽  
pp. 97-116 ◽  
Author(s):  
Arash Ghaani Farashahi
Keyword(s):  
Invariant Measure ◽  
Compact Group ◽  
Homogeneous Space ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Compact Groups ◽  
Compact Homogeneous Space ◽  
Linear Representations ◽  
Function Algebras ◽  
Convolution Function

AbstractThis paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. LetGbe a compact group andHa closed subgroup ofG. Letμbe the normalizedG-invariant measure over the compact homogeneous spaceG/Hassociated with Weil's formula and. We then present a structured class of abstract linear representations of the Banach convolution function algebrasLp(G/H,μ).

Abstract measure algebras over homogeneous spaces of compact groups

2018 ◽  
Vol 29 (01) ◽  
pp. 1850005 ◽  
Author(s):  
Arash Ghaani Farashahi
Keyword(s):  
Compact Group ◽  
Systematic Study ◽  
Measure Space ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Compact Groups ◽  
Coset Space ◽  
Left Coset ◽  
Abstract Measure

This paper presents a systematic study for abstract Banach measure algebras over homogeneous spaces of compact groups. Let [Formula: see text] be a closed subgroup of a compact group [Formula: see text] and [Formula: see text] be the left coset space associated to the subgroup [Formula: see text] in [Formula: see text]. Also, let [Formula: see text] be the Banach measure space consists of all complex measures over [Formula: see text]. Then we introduce the abstract notions of convolution and involution over the Banach measure space [Formula: see text].

scholarly journals Spread out random walks on homogeneous spaces

2020 ◽  
pp. 1-35
Author(s):  
ROLAND PROHASKA
Keyword(s):  
Random Walks ◽  
Haar Measure ◽  
Limit Theorems ◽  
Homogeneous Spaces ◽  
Locally Compact Group ◽  
Quadratic Growth ◽  
Infinite Volume ◽  
Starting Position ◽  
Markov Chain Theory ◽  
Ratio Limit

Abstract A measure on a locally compact group is said to be spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the n-step distributions: they equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.

Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups

2017 ◽  
Vol 60 (1) ◽  
pp. 111-121 ◽  
Author(s):  
Arash Ghaani Farashahi
Keyword(s):  
Hilbert Space ◽  
Operator Theory ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Compact Groups ◽  
Theory Approach ◽  
Coset Space ◽  
Trace Formulas ◽  
Left Coset ◽  
Abstract Notion

AbstractThis paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. LetGbe a compact group and letHbe a closed subgroup ofG. LetG/Hbe the left coset space ofHinGand letμbe the normalized G-invariant measure onG-Hassociated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert spaceL2(G/H,μ).

Topological Reflection Groups

1980 ◽  
Vol 32 (2) ◽  
pp. 294-309
Author(s):  
Dragomir Ž. Djoković
Keyword(s):  
Closed Subgroup ◽  
Reflection Group ◽  
Diagonal Matrix ◽  
Analogous Problem ◽  
Compact Groups ◽  
Reflection Groups ◽  
The One

Let G be a closed subgroup of one of the classical compact groups 0(n), U(n), Sp(n). By a reflection we mean a matrix in one of these groups which is conjugate to the diagonal matrix diag (–1, 1, …, 1). We say that G is a topological reflection group (t.r.g.) if the subgroup of G generated by all reflections in G is dense in G.It was shown recently by Eaton and Perlman [5] that, in case of 0(n), the whole group 0(n) is the unique infinite irreducible t.r.g. In this paper we solve the analogous problem for U(n) and Spin). Our method of proof is quite different from the one used in [5]. We treat simultaneously all the three cases.

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