Semi-groups of measures on Lie groups

Abstract

This paper grew out of discussions with S. Bochner.It would be hard now to disengage his contributions from mine.We shall characterize those families (pt)o<t<«, of finite positive measures on a Lie group Q which are weakly continuous and form a semi-group under convolution.No generality is lost in assuming that the pt are probability measures.Paul Levy [3] obtained a characterization when Q is 'R., the additive group of reals: The characteristic function <pt(^)=fcii"pt(d<r) has the form (1) <pt(t) = exp iitai, -lb? + I f (e* -1-*-^JG(do-)l where b is non-negative and G is a positive measure on <r\ such that G(0) =0 and f<r2G(da) is finite.Let us translate this result.The definition Stf-(t) = ff(r+a)pt(da) yields a semi-group (St) of transformations which are defined at least for bounded continuous functions.It is easy to prove that the limit Mf-(r) = lim-[<>,/• (r)-/(r)] t\o t(2) = af (r) + bf"(r) + J [f(r + *)-f(r) -^-jG(d<r) exists if/ is the Fourier transform of a function vanishing rapidly at infinity.Moreover, M determines the family (pt).This second form extends to a semi-group (pt) on a Lie group Q.Let us first assume(3) lim pt(E) = 1, E a neighborhood of t,(here e is the neutral element).Then the pt define a semi-group (St) of transformations on a Banach space J of functions with the property that S(f->f as / decreases to 0. Thus (St) has an infinitesimal generator; one would expect it to have an expression like (2).