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Markov semi-groups associated with the complex unimodular group \(\mathrm{Sl}(2,{\mathbb{C}})\). (English) Zbl 1422.60012

Summary: In this paper, we derive the explicit expressions of the Markov semi-groups constructed by P. Biane [“Entrelacements de semi-groupes provenant de paires de Gelfand”, ESAIM: Probab. Stat. 15, S2–S10 (2011; doi:10.1051/ps/2010025)] from the restriction of a particular positive definite function on the complex unimodular group \(\mathrm{SL}(2,{\mathbb{C}})\) to two commutative subalgebras of its universal \(C^{\star }\)-algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index \(-1\), and yield absolutely-convergent double series representations of the semi-group densities. We also supply some arguments supporting the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation \(Mp(4,{\mathbb{R}})\) and to the Landau operator in the complex plane.

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G51 Processes with independent increments; Lévy processes
42A82 Positive definite functions in one variable harmonic analysis
22E46 Semisimple Lie groups and their representations
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