Convolution of Probability Measures on Lie Groups and Homogenous Spaces

Abstract

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.2010 Mathematics Subject Classification 60B15.

Convolution Semigroups of Probability Measures on Gelfand Pairs, Revisited

“…Let M K (G) be the space of all right K-invariant Radon probability measures on G. We say that a continuous convolution semigroup (µ t , t ≥ 0) is right K-invariant, if µ t ∈ M K (G) for all t ≥ 0. In that case, µ 0 is normalised Haar measure on K, and then µ t is K-biinvariant for all t ≥ 0, as is shown in Proposition 2.1 of [20]. Indeed since for all t ≥ 0, µ t = µ 0 * µ t , left K-invariance of µ t follows from that of µ 0 .…”

“…So (µ t , t ≥ 0) is a continuous right K invariant convolution semigroup, with µ 0 being normalised Haar measure on K. Hence, by Proposition 2.1 of [20], µ t is K-bi-invariant for all t > 0.…”

“…In section 6 of [5] an attempt was made to use the generalised spherical transform to obtain a Lévy-Khintchine formula for right K-invariant convolution semigroups, in the mistaken belief that there were non-trivial elements in that class that were not K-bi-invariant. The work of [20], as described in section 2 above, shows that this was erroneous.…”

“…In the last part of the paper, an attempt was made to project this to non-compact symmetric spaces when the convolution semigroup comprises measures that are only right K-invariant. Since then Liao has shown [20] that all right K-invariant convolution semigroups are in fact Kbi-invariant. In the current paper, we will ask the question -are there any natural classes of convolution semigroups, other than the K-bi-invariant ones, that give rise to interesting classes of Markov processes on N ?…”

Transition Densities and Traces for Invariant Feller Processes on Compact Symmetric Spaces

“…It is shown in [26] that a convolution semigroup is K-left-invariant if and only if it is K-right-invariant if and only if it is K-bi-invariant. Then µ 0 = m K , (P t , t ≥ 0) as defined above 1 is a contraction semigroup on L 2 (K\G/K), and for each π ∈ G s , ( µ t (φ π ), t ≥ 0) is a strongly continuous one-parameter contraction semigroup of complex numbers.…”