Some \(L^{2}\) properties of semigroups of measures on Lie groups
Let \(\mathbb{G}\) be a locally compact group (here in most cases a compact connected Lie group), let \(\left(\mu_t\right)\) be a continuous convolution semigroup of probabilities with generator \(\mathcal{L}\) given in its Lévy-Khinchin-Hunt representation. Of particular interest is the case of symmetric \(\mu_t\) (hence self-adjoint convolution operators \(T_t\) on \(L^2\)) and their infinitesimal generators. The convolution operators \(T_t, t>0,\) are of trace class (equivalently: Hilbert-Schmidt) iff \(\mu_t\) possesses \(L^2-\)densities for all \(t>0\). (Note that on compact groups we have \(L^2 \star L^2 (\mathbb{G}) \subseteq C(\mathbb{G})\) hence \(\mu_t\) has \(L^2-\)densities for all \(t>0\) iff \(\mu_t\) has continuous densities for all \(t>0\).) The author investigates examples on a \(d-\)dimensional torus and on compact symmetric Riemannian pairs. Reviewer's remark: The interested reader will find a long list of publications concerned with densities of \(\mu_t\) and their properties, with emphasis on Gaussian laws and densities. A few examples (far from being complete): [\textit{E. Siebert}, Manuscr. Math. 14, 41--55 (1974; Zbl 0293.60013);\ Math. Ann. 210, 129--147 (1974; Zbl 0273.43003); Proc. Am. Math. Soc. 91, 298--305 (1984; Zbl 0548.60008); Monatsh. Math. 93, 239--253 (1982; Zbl 0477.60008)] [\textit{Ch. Berg}, Invent. Math. 32, 49--100 (1976; Zbl 0371.31007)] [\textit{A. Janssen}, Math. Ann. 246, 233--240 (1980; Zbl 0407.60005);\ Arch. Math. 38, 565--576 (1982; Zbl 0476.60005)] [\textit{A. Bendikov} and \textit{L. Saloff Coste}, Contemp. Math. 338, 41--63 (2003; Zbl 1040.60006); Heat kernels and analysis on manifolds, graphs, and metric spaces. Lecture notes from a quarter program on heat kernels, random walks, and analysis on manifolds and graphs, April 16--July 13, 2002, Paris, France. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 338, 41-63 (2003; Zbl 1040.60006), Sections 3 and 4].
square integrable densities, continuous densities
scientific article; zbMATH DE number 5614937
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