Convolution Semigroups of Probability Measures on Gelfand Pairs, Revisited

Markov kernels, convolution semigroups, and projective families of probability measures

For a measurable space (E,E ), we denote by E+ the set of functions E → [0,∞] that are E → B[0,∞] measurable. It can be proved that if I : E+ → [0,∞] is a function such that (i) f = 0 implies that I(f) = 0, (ii) if f, g ∈ E+ and a, b ≥ 0 then I(af + bg) = aI(f) + bI(g), and (iii) if fn is a sequence in E+ that increases pointwise to an element f of E+ then I(fn) increases to I(f), then there a ...

Gelfand pairs

Let K ⊂ G be a compact subgroup of a real Lie group G. Denote by D(X) thealgebra of G-invariant differential operators on the homogeneous space X = G/K. ThenX is called commutative or the pair (G,K) is called a Gelfand pair if the algebra D(X)is commutative. Symmetric Riemannian homogeneous spaces introduced by Élie Cartanand weakly symmetric homogeneous spaces introduced by Sel...

Groups, Probability and Combinatorics: Different Aspects in Gelfand Pairs Theory

Weak Convergence of Convolution Products of Probability Measures on Semihypergroups

Let S be a topological semihypergroup. As it is known for hypergroups, the lack of an algebraic structure on a semihypergroup pause a serious challenge in extending results from semigroups. We use the notion of concretization or pseudomultiplication, to prove some results on weak convergence of the sequence of averages of convolution powers of probability measures on topological semihypergroups...

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...