CONVOLUTION HEMIGROUPS OF BOUNDED VARIATION ON A LIE PROJECTIVE GROUP

Abstract

In a previous article [11] we studied the central limit theorem for infinitesimal triangular arrays of probability measures on a Lie group. Since the work of Berg [2] and more recently of Bendikov [1] similar studies appeared to be urgent for the infinite-dimensional torus group and beyond that for the class of Lie projective groups which among others contains all compact groups. Consequently we started extending the existing theory within the enlarged framework of Lie projective groups. In the present contribution continuous convolution hemigroups (μ(s, t)) of probability measures on a Lie projective group G are investigated with respect to generation, representation and occurrence as limits of non-commutative infinitesimal triangular arrays.

The layout of our exposition is as follows. In Section 2 some facts on Lie projective groups G with Lie algebra [Lscr ](G), Lie system [Hfr ], projective basis (X_i)_i∈I and corresponding projective weak coordinate system (x_i)_i∈I are collected to make the reader familiar with the setting. Section 3 contains the basic methodical result yielding the intended generalisations. It is shown in Proposition 3.3 that a hemigroup (μ(s, t)) in the set [Mfr ]¹(G) of probability measures on G corresponds to a triplet (a, B, η) in the set ℙ_bv(ℝ₊, G) of characteristics of bounded variation if and only if for each H∈[Hfr ] the projection (p_H(μ(s, t))) in [Mfr ]¹(G/H) corresponds to a certain triplet (a^H, B^H, η^H) in the set ℙ_bv(ℝ₊, G/H). Here the correspondence between hemigroups and triplets is achieved via weak backward evolution equations with respect to (X_i)_i∈I and (x_i)_i∈I (see Definition 3.2). Applying the reduction method based on this result, the convergence of infinitesimal arrays {μ_{n[lscr ]}[ratio ](n,[lscr ])∈ℕ²} in [Mfr ]¹(G) towards hemigroups (μ(s, t)) of continuous weak bounded variation is proved in Section 4. In fact, sufficient conditions involving the triplet (a, B, η) are given in order that the sequence (μ_n (s, t))_{n[ges ]1} of ‘scaled’ convolution products μ_{n, kn(s)+1}[midast ]…[midast ] μ_{n, kn(t)} converges weakly to (μ(s, t)). Moreover, necessary and sufficient conditions are given for the limiting hemigroup to be a diffusion hemigroup. Section 5 is devoted to the existence and uniqueness results leading to the one-to-one correspondence between the set ℙ_bv(ℝ₊, G) and the set of hemigroups of continuous weak bounded variation on G. In Section 6 we illustrate the technique of Lie projectivity of the infinite-dimensional torus group and of the p-adic solenoidal group. Moreover, we sketch a little-known example of a hemigroup appearing in atomic physics and derive from it some thoughts towards a perturbation theory for hemigroups.