Infinitely divisible central probability measures on compact Lie groups-regularity, semigroups and transition kernels
The investigation of central probabilities on locally compact abelian groups has a long history, with applications to probability theory as well as to analysis. E.g., the existence of central Gaussian laws has a strong impact on the underlying group, see e.g., the nice characterization of central groups by \textit{E. Siebert} [``Absolut-Stetigkeit und Träger von Gauß-Verteilungen auf lokalkompakten Gruppen'', Math. Ann. 210, 129--147 (1974; Zbl 0273.43003)]. Hence groups admitting a central Gaussian semigroup with full support are (connected) and central, in particular, MAP groups. The author considers central continuous convolution semigroups on \textit{compact} Lie groups, which behave similarly to the heat semigroup. More precisely, he induces a class \(\mathrm{CID}_{\mathbb{R}}\) chracterized by Fourier transforms representable as \(\widehat{\mu}(\pi) = e^{-\eta\left(\chi_\pi^{1/2}\right)}\cdot I_\pi\), where \(\eta \) denotes a second characteristic function of an infinitely divisible law on \(\mathbb{R}\), \(\pi\) denoting an irreducible representation. Various examples arise by subordination of symmetric central Gaussian laws. For the background on continuous convolution semigroups the reader is referred, e.g., to the monographs of \textit{H. Heyer} [Probability measures on locally compact groups. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0376.60002)], and of the author [Lévy processes and stochastic calculus. 2nd ed. Cambridge: Cambridge University Press (2009; Zbl 1200.60001)]. Section 3 is concerned with regularity (\(L^2\) integrability, continuity or smoothness) of densities expressible as series \(k(\sigma) =\sum_\pi d_\pi\overline{c}_\pi\chi_\pi(\sigma)\) for almost all \(\sigma\in G\), where the Fourier transform is given as \(\widehat{\mu}(\pi)=c_\pi I_\pi \), and \(\chi_\pi\) denotes the character of \(\pi\). In analogy to the abelian case, regularity properties of the density \(k\) are described by regularity of the series \(\sum d_\pi^2|c_\pi|\). The following section 4 is concerned with the corresponding \(C_0\)-semi{\-}groups \((T_t)\) of convolution operators acting on \(L^2\) and their infinitesimal generators. These operators are represented as pseudo differential operators, acting on Sobolev spaces \(\mathcal{H}_p\). For central convolution semigroups \((\mu_t)\), with densities \(k_t\), the \textit{transition density} is defined as \(h_t(\sigma, \tau) := k_t(\sigma^{-1}\tau)\). If \(k_t\) is continuous, also series representations for \(h_t\) are obtained. The operators \(T_t\) are of trace-class and the traces are calculated by the densities, where traces are considered on the Hilbert space \(L^2\) as well as on the subspace \(L^2_c\) of central functions. Finally, in section 6, the author investigates (for compact semisimple Lie groups) the behaviour of (continuous) densities for small times \(t\). Throughout, all obtained results are explained by examples \((\mu_t)\), in particular, central Gaussian laws, and their subordinated laws, e.g., Laplace laws, stable laws and relativistic Schrödinger laws.
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