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Some Martingales Associated With Multivariate Bessel Processes

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Abstract

We study Bessel processes on Weyl chambers of types A and B on \(\mathbb{R}^N\). Using elementary symmetric functions, we present several space-timeharmonic functions and thus martingales for these processes \((X_t)_{t\ge0}\)which are independent from one parameter of these processes. As a consequence, \(p_t(y):= \mathbb{E}(\prod_{i=1}^N (y-X_t^i))\) can be expressed via classical orthogonal polynomials. Such formulas on characteristic polynomials admit interpretations in random matrix theory where they are partially known by Diaconis, Forrester, and Gamburd.

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Acknowledgements

The authors would like to thank the anonymous referee for several useful hints.

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Authors and Affiliations

  1. Department of Probability Theory and Statistics, Eötvös Loránd University, Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary

    M. Kornyik

  2. Department of Quantum Optics and Quantum Information, Wigner Research Centre for Physics, Konkoly-Thege Miklós út 29-33, H-1121, Budapest, Hungary

    M. Kornyik

  3. Fakultät Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44221, Dortmund, Germany

    M. Voit & J. Woerner

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  1. M. Kornyik
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  2. M. Voit
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  3. J. Woerner
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Correspondence to M. Voit.

Additional information

The first author has been supported by the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131 High-dimensional Phenomena in Probability – Fluctuations and Discontinuity to visit Dortmund for the preparation of this paper, and also by Project no. ED 18-1-2019-0030 (Application domain specific highly reliable IT solutions subprogramme) which has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the Thematic Excellence Programme funding scheme.

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Kornyik, M., Voit, M. & Woerner, J. Some Martingales Associated With Multivariate Bessel Processes. Acta Math. Hungar. 163, 194–212 (2021). https://doi.org/10.1007/s10474-020-01096-5

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  • DOI: https://doi.org/10.1007/s10474-020-01096-5

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