Probability on Compact Lie Groups

Convolution roots and embeddings of probability measures on Lie groups

We show that for a large class of connected Lie groups G, viz. from class C described below, given a probability measure μ on G and a natural number n, for any sequence {νi} of n th convolution roots of μ there exists a sequence {zi} of elements of G, centralising the support of μ, and such that {ziνiz i } is relatively compact; thus the set of roots is relatively compact ’modulo’ the conjugati...

Probability of Mutually Commuting N-Tuples in Some Classes of Compact Groups

Lattice of full soft Lie algebra

In ‎this ‎paper, ‎we ‎study ‎the ‎relation ‎between ‎the ‎soft ‎sets ‎and ‎soft ‎Lie ‎algebras ‎with ‎the ‎lattice theory. ‎We ‎introduce ‎the ‎concepts ‎of ‎the ‎lattice ‎of ‎soft ‎sets, ‎full ‎soft ‎sets ‎and ‎soft ‎Lie ‎algebras ‎and next, we ‎verify ‎some ‎properties ‎of ‎them. We ‎prove ‎that ‎the ‎lattice ‎of ‎the ‎soft ‎sets ‎on ‎a fixed parameter set is isomorphic to the power set of a ...

Brownian Processes for Monte Carlo Integration on Compact Lie Groups

This paper proposes a Monte Carlo approach for the evaluation of integrals of smooth functions defined on compact Lie groups. The approach is based on the ergodic property of Brownian processes in compact Lie groups. The paper provides an elementary proof of this property and obtains the following results. It gives the rate of almost sure convergence of time averages along with a “large deviati...

Pro-Lie Groups: A Survey with Open Problems

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity compo...