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Infinitely divisible central probability measures on compact Lie groups-regularity, semigroups and transition kernels. (English) Zbl 1237.60005
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 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 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 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 \(C_0\)-semigroups \((T_t)\) 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 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.

MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G51 Processes with independent increments; Lévy processes
47D07 Markov semigroups and applications to diffusion processes
43A05 Measures on groups and semigroups, etc.
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[1] Applebaum, D. (2001). Lévy processes in stochastic differential geometry. In Lévy Processes : Theory and Applications (O. Barndorff-Nielsen, T. Mikosch and S. Resnick, eds.) 111-137. Birkhäuser, Boston, MA. · Zbl 0984.60056
[2] Applebaum, D. (2001). Operator-valued stochastic differential equations arising from unitary group representations. J. Theoret. Probab. 14 61-76. · Zbl 0979.60059 · doi:10.1023/A:1007816930696
[3] Applebaum, D. (2008). Probability measures on compact groups which have square-integrable densities. Bull. Lond. Math. Soc. 40 1038-1044. · Zbl 1163.60003 · doi:10.1112/blms/bdn088
[4] Applebaum, D. (2009). Lévy Processes and Stochastic Calculus , 2nd ed. Cambridge Studies in Advanced Mathematics 116 . Cambridge Univ. Press, Cambridge. · Zbl 1200.60001
[5] Applebaum, D. (2009). Some L 2 properties of semigroups of measures on Lie groups. Semigroup Forum 79 217-228. · Zbl 1177.43002 · doi:10.1007/s00233-008-9130-0
[6] Berg, C. and Forst, G. (1975). Potential Theory on Locally Compact Abelian Groups . Springer, New York. · Zbl 0308.31001
[7] Blumenthal, R. M. and Getoor, R. K. (1959). The asymptotic distribution of the eigenvalues for a class of Markov operators. Pacific J. Math. 9 399-408. · Zbl 0086.33901 · doi:10.2140/pjm.1959.9.399
[8] Blumenthal, R. M. and Getoor, R. K. (1960). Some theorems on stable processes. Trans. Amer. Math. Soc. 95 263-273. · Zbl 0107.12401 · doi:10.2307/1993291
[9] Born, È. (1989). An explicit Lévy-Hinčin formula for convolution semigroups on locally compact groups. J. Theoret. Probab. 2 325-342. · Zbl 0678.60010 · doi:10.1007/BF01054020
[10] Elworthy, D. (1988). Geometric aspects of diffusions on manifolds. In École d’Été de Probabilités de Saint-Flour XV-XVII , 1985 - 87. Lecture Notes in Math. 1362 277-425. Springer, Berlin. · Zbl 0658.58040
[11] Faraut, J. (2008). Analysis on Lie Groups : An Introduction. Cambridge Studies in Advanced Mathematics 110 . Cambridge Univ. Press, Cambridge. · Zbl 1147.22001
[12] Fegan, H. D. (1978). The heat equation and modular forms. J. Differential Geom. 13 589-602 (1979). · Zbl 0437.22010
[13] Fegan, H. D. (1983). The fundamental solution of the heat equation on a compact Lie group. J. Differential Geom. 18 659-668 (1984). · Zbl 0596.35052
[14] Fegan, H. D. (1991). Introduction to Compact Lie Groups. Series in Pure Mathematics 13 . World Scientific, River Edge, NJ. · Zbl 0743.22002
[15] Getoor, R. K. (1959). Markov operators and their associated semi-groups. Pacific J. Math. 9 449-472. · Zbl 0086.33804 · doi:10.2140/pjm.1959.9.449
[16] Hare, K. E. (1998). The size of characters of compact Lie groups. Studia Math. 129 1-18. · Zbl 0946.43006 · eudml:216489
[17] Heyer, H. (1968). L’analyse de Fourier non-commutative et applications à la théorie des probabilités. Ann. Inst. H. Poincaré Sect. B ( N.S. ) 4 143-164. · Zbl 0165.19102 · numdam:AIHPB_1968__4_2_143_0 · eudml:76880
[18] Heyer, H. (1972). Infinitely divisible probability measures on compact groups. In Lectures on Operator Algebras. Lecture Notes in Math. 247 55-249. Springer, Berlin. · Zbl 0243.60010
[19] Heyer, H. (1977). Probability Measures on Locally Compact Groups . Springer, Berlin. · Zbl 0376.60002
[20] Hunt, G. A. (1956). Semi-groups of measures on Lie groups. Trans. Amer. Math. Soc. 81 264-293. · Zbl 0073.12402 · doi:10.2307/1992917
[21] Ichinose, T. (1989). Essential selfadjointness of the Weyl quantized relativistic Hamiltonian. Ann. Inst. H. Poincaré Phys. Théor. 51 265-297. · Zbl 0721.35059 · numdam:AIHPA_1989__51_3_265_0 · eudml:76468
[22] Jacob, N. (1996). Pseudo-Differential Operators and Markov Processes. Mathematical Research 94 . Akademie Verlag, Berlin. · Zbl 0860.60002
[23] Jacob, N. (2005). Pseudo-Differential Operators and Markov Processes : Vol. III. Markov Processes and Applications . Imperial College Press, London. · Zbl 1076.60003 · ebooks.worldscinet.com
[24] Kim, P. T. and Richards, D. S. (2001). Deconvolution density estimators on compact Lie groups. Contemp. Math. 287 155-171. · Zbl 1020.62029
[25] Knapp, A. W. (2002). Lie Groups Beyond an Introduction , 2nd ed. Progress in Mathematics 140 . Birkhäuser, Boston, MA. · Zbl 1075.22501
[26] Koo, J.-Y. and Kim, P. T. (2008). Asymptotic minimax bounds for stochastic deconvolution over groups. IEEE Trans. Inform. Theory 54 289-298. · Zbl 1304.94012 · doi:10.1109/TIT.2007.911263
[27] Kunita, H. (1999). Analyticity and injectivity of convolution semigroups on Lie groups. J. Funct. Anal. 165 80-100. · Zbl 0957.60008 · doi:10.1006/jfan.1999.3403
[28] Liao, M. (2004). Lévy processes and Fourier analysis on compact Lie groups. Ann. Probab. 32 1553-1573. · Zbl 1054.60003 · doi:10.1214/009117904000000306
[29] Liao, M. (2004). Lévy Processes in Lie Groups. Cambridge Tracts in Mathematics 162 . Cambridge Univ. Press, Cambridge. · Zbl 1076.60004
[30] Lo, J. T. H. and Ng, S. K. (1988). Characterizing Fourier series representation of probability distributions on compact Lie groups. SIAM J. Appl. Math. 48 222-228. · Zbl 0639.60009 · doi:10.1137/0148011
[31] Ragozin, D. L. (1972). Central measures on compact simple Lie groups. J. Funct. Anal. 10 212-229. · Zbl 0286.43002 · doi:10.1016/0022-1236(72)90050-X
[32] Rosenberg, S. (1997). The Laplacian on a Riemannian Manifold : An Introduction to Analysis on Manifolds. London Mathematical Society Student Texts 31 . Cambridge Univ. Press, Cambridge. · Zbl 0868.58074
[33] Ruzhansky, M. and Turunen, V. (2010). Pseudo-differential Operators and Symmetries : Background Analysis and Advanced Topics. Pseudo-Differential Operators. Theory and Applications 2 . Birkhäuser, Basel. · Zbl 1193.35261 · doi:10.1007/978-3-7643-8514-9
[34] Said, S., Lageman, C., LeBihan, N. and Manton, J. H. (2010). Decompounding on compact Lie groups. IEEE Trans. Inf. Theory 56 2766-2777. · Zbl 1366.94123 · doi:10.1109/TIT.2010.2046216
[35] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68 . Cambridge Univ. Press, Cambridge. · Zbl 0973.60001
[36] Siebert, E. (1981). Fourier analysis and limit theorems for convolution semigroups on a locally compact group. Adv. in Math. 39 111-154. · Zbl 0469.60014 · doi:10.1016/0001-8708(81)90026-8
[37] Sugiura, M. (1971). Fourier series of smooth functions on compact Lie groups. Osaka J. Math. 8 33-47. · Zbl 0223.43006
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