Convolution Semigroups of Measures

Abstract

Properties of convolution semigroups of probability measures and associated Hunt semigroups of operators on general Lie groups are described. We outline a proof of Hunt’s theorem which gives a Lévy-Khintchine type representation for the infinitesimal generator of such semigroups. Then we study $L^{2}$-properties of these semigroups, specifically finding conditions which guarantee self-adjointness, injectivity, analyticity and the trace-class property. We then use the Fourier transform on a compact Lie group to find a direct analogue of the Lévy-Khintchine formula. We prove the central limit theorem for Gaussian semigroups of measures. Then we describe the technique of subordination, and discuss regularity of densities for some semigroups of measures that are obtained by subordinating the standard Gaussian. Finally we investigate the small-time asymptotic behaviour of some subordinated densities.