Erdal Coskun

Abstract

        A convolution semigroup plays an important role in the theory of probability measure on Lie groups. The basic problem is that one wants to express a semigroup as a Lèvy-Khinchine formula. If (μ_t)_{t Î R+*}   is a continuous semigroup of probability measures on a Hilbert-Lie group G, then we define  T_{μ t} f := ∫ f_a μ_t (da)  (f Î C_*(G);  t > 0). It is apparent that (T_{μ t} )_{t Î R+*} is a continuous operator semigroup on the space C_*(G) with the infinitesimal generator N. The generating functional A of this semigroup is defined by Af := \lim\limits _t↓0 1/t (T_{μ t}  f(e) - f(e)). We consider the problem of construction of a subspace C₂.(G) of C_*(G) such that the generating functional A on C₂.(G) exists. This result will be used later to show that Lèvy-Khinchine formula holds for Hilbert-Lie groups