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Publications

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List of Publications

Uwe Franz

Preprints

Books

  • Stochastic processes and Operator Calculus on Quantum Groups, (with R. Schott), Kluwer Academic Publishers (now Springer Verlag), Mathematics and Its Applications
    Vol. 490, ISBN 978-90-481-5290-2 (Print), 978-94-015-9277-2 (Online), DOI 10.1007/978-94-015-9277-2, 236 pp., July 1999.

Books edited

  • Ph. Biane, L. Bouten, F. Cipriani, N. Konno, N. Privault, and Q. Xu, Quantum Potential Theory, (U. Franz and M. Schürmann, eds.), Lecture Notes of a School held in Greifswald in Greifswald in Februar-March 2007, Springer Lecture Notes in Mathematics, Vol. 1954, 2008, DOI 10.1007/978-3-540-69365-9.
  • O.E. Barndorff-Nielsen, U. Franz, R. Gohm, B. Kümmerer, S. Thorbjørnsen, Quantum Independent Increment Processes II : Quantum Lévy Processes, Classical Probability, and Applications, (U. Franz and M. Schürmann, eds.), Lecture Notes of a School held in Greifswald in Greifswald in March 2003, Springer Lecture Notes in Mathematics, Vol. 1866, 2006, DOI 10.1007/11376637.
  • D. Applebaum, B.V.R. Bhat, J. Kustermans, J.M. Lindsay, Quantum Independent Increment Processes I : From Classical Probability to Quantum Stochastic Calculus, (U. Franz and M. Schürmann, eds.), Lecture Notes of a School held in Greifswald in Greifswald in March 2003, Springer Lecture Notes in Mathematics, Vol. 1865, 2005, DOI 10.1007/b105131.

Articles in Journals

  • Second cohomology groups of the Hopf*-algebras associated to universal unitary quantum groups, (with Biswarup Das, Anna Kula and Adam Skalski), Preprint arXiv:2104.07933, April 2021, acccepted for publication in the Annales de l’Institut Fourier.
  • The RFD and Kac quotients of the Hopf*-algebras of universal orthogonal quantum groups, (with Biswarup Das and Adam Skalski), Preprint arXiv:2012.00416, December 2020, accepted for publication in Les Annales Mathématiques Blaise Pascal.
  • Invariant Markov semigroups on quantum homogeneous spaces, with Biswarup Das and Xumin Wang, Journal of Noncommutative Geometry 15 (2), 531-580 (2021), DOI 10.4171/JNCG/404.
  • Beurling-Fourier algebras of compact quantum groups : characters and finite dimensional representations, with Hunhee Lee, Indiana Univ. Math. J. 70 No. 2 (2021), 605–637, DOI 10.1512/iumj.2021.70.8405.
  • Lévy-Khintchine decompositions for generating functionals on universal CQG-algebras, (with Biswarup Das, Anna Kula, and Adam Skalski), Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 21, No. 03, 1850017 (2018) DOI 10.1142/S0219025718500170
  • Integration over the quantum diagonal subgroup and associated Fourier-like algebras, (with Hun Hee Lee and Adam Skalski), International Journal of Mathematics Vol. 27, 1650073 (2016) [37 pages], 10.1142/S0129167X16500737.
  • Conic structure of the non-negative operator convex functions on (0, ∞), (with F. Hiai), Ann. Funct. Anal. 5, 158-175, 2014.
  • Higher order extension of Löwner’s theory : Operator k-tone functions, (with F. Hiai, É. Ricard), Trans. Amer. Math. Soc. 366, 3043-3074, 2014.
  • Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory, (with F. Cipriani, A. Kula), J. Funct. Anal. 266, 2789-294, 2014.
  • Classication of Idempotent States on the compact quantum groups U q (2), SU q (2), and SO q (3), (with A. Skalski, R. Tomatsu), J. Noncom. Geom. 7, 221-254, 2013.
  • On square roots of the Haar state on compact quantum groups, (with A. Skalski and R. Tomatsu), J. Pure and Applied Alg. 216, 2079-2093, 2012.
  • Idempotent states and the inner linearity property, (with T. Banica and A. Skalski),
    Bulletin Polish Academy Sciences Mathematics, vol. 60, 123-132, 2012.
  • Piecewise deterministic Markov processes as limits of Markov jump processes, (with V. Liebscher und S. Zeiser), Adv. In Appl. Proabab. 44 (2012), 729-748.
  • Computing Coordinates on Lie Groups, (with Ph. Feinsilver and R. Schott) to appear in International Journal of Pure and Applied Mathematics, vol. 60, no. 4, 2010.
  • On idempotent states on quantum groups, (with A. Skalski), Journal of Algebra, Vol. 322, no. 5, pp. 1774-1802, 2009.
  • Monotone and Boolean Convolutions for Non-compactly Supported Probability Measures, Indiana University Mathematics Journal, Vol. 58, no. 3, pp. 1151-1186, 2009.
  • Autocatalytic genetic networks modeled by piecewise-deterministic Markov processes, (with Z. Zeiser, V. Liebscher), Journal of Mathematical Biology, Vol. 60, no. 2, pp. 207-246, 2009.
  • Hybrid Modeling of Noise Reduction by a Negatively Autoregulated System, with S.Zeiser, J. Müller, V. Liebscher, Bulletin of Mathematical Biology 71, 1006-1024, 2009.
  • Boolean convolution of probability measures on the unit circle, in ``Analyse et probabilités’’, P. Biane, J. Faraut, and H. Ouerdiane (eds.), Séminaires et Congrès 16, 83-93, 2009.
  • Simulation of genetic networks modelled by piecewise deterministic Markov processes, with S. Zeiser, O. Wittich, V. Liebscher, IET Syst. Biol. 2, no. 3, 112-1135, 2008.
  • On ergodic properties of convolution operators associated with compact quantum groups, (with A.\ Skalski), Colloq. Math. 113, no. 1, 13-23, 2008.
  • Approximation of quantum Lévy processes by quantum random walks, (with A. Skalski), Proc. Indian Acad. Sci. Math. Sci. 118, no. 2, 281-288, 2008.
  • Multiplicative monotone convolutions, in "Quantum Probability’’, Marek Bozejko, Wojciech Mlotkowski and Janusz Wysoczanski (eds.), Banach Center Publications Volume 73, p. 153-166, 2006.
  • Quasi-Invariance Formulas for Components of Quantum Lévy Processes, (with N. Privault), Inf. Dim. Anal., Quantum Prob. and Rel. Topics, Vol. 7, No. 1, pp. 131-145, 2004.
  • The Affine Group and Generalized Gegenbauer Polynomials, (with Ph. Feinsilver) Computers and Mathematics with Applications Vol. 41, 1173-1182, 2001.
  • Solving Evolution Equations on Lie Groups, (with Ph. Feinsilver and R. Schott), Journal of Physics A : Math. and Gen. 33, pp. 2777-2798, 2000.
  • Limit Theorems for the Hierarchy of Freeness, (with R. Lenczewski), Probab. Math. Stat. Vol. 19, pp. 23-41, 1999.
  • Phase Retrieval for Probability Distributions on Quantum Groups and Braided Groups, (with D. Neuenschwander and R. Schott), Journal of Theoretical Probability, Vol. 12, No. 2, pp. 585-594, 1999.
  • Diffusions on Braided Spaces, (with R. Schott), Journal of Math. Phys., Vol. 39, No. 5, pp. 2748-2762, 1998.
  • Evolution Equations and Lévy Processes on Quantum Groups, (with R. Schott), Journal of Physics A : Math. and Gen. 31, pp. 1395-1404, 1998.
  • Gauss Laws in the Sense of Bernstein and Uniqueness of Embedding into Convolution Semigroups on Quantum Groups and Braided Groups, (with D. Neuenschwander and R. Schott), Probab. Theory Relat. Fields 109, pp. 101-127, 1997.
  • Duality and Multiplicative Processes on Quantum Groups, (with Ph. Feinsilver and R. Schott), Journal of Theoretical Probability, Vol. 10, No. 3, pp. 795-818, 1997.

Communications to the Comptes Rendus de l’Académie des Sciences Paris

  • A New Characterisation of Idempotent States on Finite and Compact Quantum Groups, (with A. Skalski), Comptes Rendus Mathematique, Vol. 347, Issue 17-18, pp. 991-996, 2009.
  • Smoothness of Wigner densities on the affine algebra, (with N. Privault and R. Schott), Comptes Rendus Mathématiques, Vol. 337, Issue 9, pp. 609-614, 2003.
  • Malliavin Calculus for Quantum Stochastic Processes, (with R. Léandre and R. Schott), C. R. Acad. Sci. Paris, t. 328, Série I, p. 1061-1066, 1999.
  • Gauss Laws in the Sense of Bernstein and Uniqueness of Embedding into Covolution Semigroups on Quantum Groups and Braided Groups, (with D. Neuenschwander and R. Schott), C. R. Acad. Sci. Paris, t. 324, Série I, p. 827-832, 1997.
  • Feynman-Kac Formula and Appell Systems on Quantum Groups, (with Ph. Feinsilver and R. Schott), C. R. Acad. Sci. Paris, t. 321, Série I, p. 1615-1619, 1995.

Conferences with Committee and Proceedings

  • Markov property of monotone Lévy processes, (with N. Muraki), in Infinite Dimensional Harmonic Analysis III, Proceedings of the Third German-Japanese Symposium, University of Tübingen, Germany, 15-20 September 2003, H. Heyer, T. Hirai, T. Kawazoe, K. Saito (Eds.), p. 37-57, 2005, see also math.PR/0401390.
  • Lévy Processes on Quantum Hypergroups, (with M. Schürmann), in Infinite Dimensional Harmonic Analysis, Transactions of a Japanese-German Symposium held from Sept. 20th to 24th, 1999, at the University of Kyoto, H. Heyer, T. Hida, and N. Obata (Eds.), p. 93-114, 2000.
  • Infinitely divisible representations of Uq(g), (with V.K. Dobrev, H.-D. Doebner, and R.Schott), Proceedings of the International Symposium Quantum Theory and Symmetries, 18-22 July 1999, Goslar, Germany, H.-D. Doebner, V.K. Dobrev, J.-D. Hennig, W. Lücke (Eds.), World Scientific, Singapore, p. 464-469, 2000.
  • Lévy Processes on Uq(g), (with V.K. Dobrev, H.-D. Doebner, and R. Schott), Proceedings of the Third International Workshop Lie Theory and its Applications in Physics III, 11-14 July 1999, Clausthal, Germany, H.-D. Doebner, V.K. Dobrev, J. Hilgert (Eds.), World Scientific, Singapore, p. 280-292, 2000.
  • Lévy Processes on Quantum Groups, in ``Probability on Algebraic Structures’’, G. Budzban, Ph. Feinsilver, and A. Mukherjea (Eds.), Contemporary Mathematics Vol. 261, American Mathematical Society, p. 161-179, 2000
  • Lévy Processes on U_q(g) as Infinitely Divisible Representations, (with V.K. Dobrev, H.-D. Doebner, and R.Schott), in ``Probability on Algebraic Structures’’, G. Budzban, Ph. Feinsilver, and A. Mukherjea (Eds.), Contemporary Mathematics Vol. 261, American Mathematical Society, p. 181-192, 2000.
  • A Mixed Classical-Quantum Limit Theorem on Bialgebras, Workshop on Quantum Probability, Gdansk, Poland, 1-6. July 1997, published in Banach Center Publications, Vol. 43, pp. 183-189, 1998.
  • On the Computations of Polynomial Representations of Nilpotent Lie Groups : A Symbolic Mathematical Approach, (with Ph. Feinsilver and R. Schott), Proceedings of the 1997 ACM Symposium on Applied Computing, ACM Publ., p. 537-539, 1997.
  • A Stochastic Approach to Evolution Equations on Nilpotent Quantum Groups, (with R. Schott), "Quantum Group Symposium at Group21"(Proceedings of the Quantum Group Symposium at the XXI International Colloquium on Group Theoretical Methods in Physics, Goslar, Germany, 15-20. July 1996), H.-D. Doebner and V.K. Dobrev (Eds.), Heron Press, Sofia, p. 152-157, 1997 ; ISBN 954-590-027-5.
  • Duality and Multiplicative Processes on Quantum Groups, (with Ph. Feinsilver and R. Schott), Proceedings of the International Conference on ``Nonlinear, Deformed and Irreversible Quantum Systems’’, held in Clausthal, Germany, August 15-19, 1994. H.D. Doebner, V.K. Dobrev, P. Nattermann, editors, World Scientific, p. 462-468, 1995.
  • Generalized Lie algebras, (with B. Gruber), in Symmetries in Science VI : From the rotation group to quantum algebras, Proceedings of a symposium held in Bregenz, Austria, August 2-7, 1992. Plenum Press, New York, p. 247-253, 1993.

Habilitation Thesis

  • The Theory of Quantum Lévy Processes, Habilitation thesis EMAU Greifswald, 204 pages, math.PR/0407488, 2004.

Ph.D. Thesis

  • Contribution to the Study of Stochastic Processes on Quantum Groups, Unversité Henri Poincaré - Nancy 1, June 1997.