Electronic Communications in Probability

A matrix Bougerol identity and the Hua-Pickrell measures

Theodoros Assiotis

Full-text: Open access

Abstract

We prove a Hermitian matrix version of Bougerol’s identity. Moreover, we construct the Hua-Pickrell measures on Hermitian matrices, as stochastic integrals with respect to a drifting Hermitian Brownian motion and with an integrand involving a conjugation by an independent, matrix analogue of the exponential of a complex Brownian motion with drift.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 7, 11 pp.

Dates
Received: 20 October 2017
Accepted: 5 January 2018
First available in Project Euclid: 21 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1519182082

Digital Object Identifier
doi:10.1214/18-ECP107

Mathematical Reviews number (MathSciNet)
MR3771765

Zentralblatt MATH identifier
1398.60010

Subjects
Primary: 60G

Keywords
random matrices stochastic processes integrable probability

Rights
Creative Commons Attribution 4.0 International License.

Citation

Assiotis, Theodoros. A matrix Bougerol identity and the Hua-Pickrell measures. Electron. Commun. Probab. 23 (2018), paper no. 7, 11 pp. doi:10.1214/18-ECP107. https://projecteuclid.org/euclid.ecp/1519182082


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References

  • [1] L. Alili, D. Dufresne, M. Yor, Sur l’identite de Bougerol pour les fonctionnelles exponentielles du mouvement brownien avec drift. In Exponential Functionals and Principal Values related to Brownian Motion. A collection of research papers; Biblioteca de la Revista Matematica, IberoAmericana, ed. M. Yor, 3–14, (1997).
  • [2] T. Assiotis, Hua-Pickrell diffusions and Feller processes on the boundary of the graph of spectra, Available from https://arxiv.org/abs/1703.01813, (2017).
  • [3] P. Baldi, E. Casadio Tarabusi, A. Figa-Talamanca, M. Yor, Non-symmetric hitting distributions on the hyperbolic half-plane and subordinated perpetuities, Revista Matematica IberoAmericana, 17, Issue 1, 587–605, (2001).
  • [4] J. Bertoin, D. Dufresne, M. Yor, Some two-dimensional extensions of Bougerol’s identity in law for the exponential functional of linear Brownian motion, Revista Matematica IberoAmericana, 29, Issue 4, 1307–1324, (2013).
  • [5] A. Borodin, G. Olshanski, Infinite Random Matrices and Ergodic Measures, Communications in Mathematical Physics, Vol. 223, Issue 1, 87–123 (2001).
  • [6] P. Bougerol, Exemples de theoremes locaux sur les groupes resolubles, Annales de l’ Institut Henri Poincare, 19, 369–391, (1983).
  • [7] P. Bougerol, Personal communication.
  • [8] A. Bufetov, Y. Qiu, The explicit formulae for scaling limits in the ergodic decomposition of infinite Pickrell measures, Arkiv for Matematik, Vol. 54, Issue 2, 403–435, (2016).
  • [9] M-F. Bru, Wishart processes, Journal of Theoretical Probability, Vol. 4, Issue 4, 725–751, (1991).
  • [10] Y. Doumerc, PhD Thesis: Matrices aleatoires, processus stochastiques et groupes de reflexions, Available from http://perso.math.univ-toulouse.fr/ledoux/files/2013/11/PhD-thesis.pdf, (2005).
  • [11] D. Dufresne, The distribution of a perpetuity, with application to risk theory and pension funding, Scandinavian Actuarial Journal, no. 1, 39–79, (1990).
  • [12] J. Franchi, Y. Le Jan, Hyperbolic dynamics and Brownian motion: An introduction, Oxford Mathematical Monographs, (2012).
  • [13] P. Graczyk, J. Malecki Multidimensional Yamada-Watanabe theorem and its applications to particle systems, Journal of Mathematical Physics, Volume 54, Issue 2, (2013).
  • [14] P. Graczyk, J. Malecki, Strong solutions of non-colliding particle systems, Electronic Journal of Probability, Vol. 19, 1–21, (2014).
  • [15] Hua L.K., Harmonic analysis of functions of several complex variables in the classical domains, Chinese edition: Peking, Science Press (1958), English edition: Transl. Math. Monographs 6, RI Providence, American Mathematical Society (1963).
  • [16] Y. Neretin, Hua type integrals over unitary groups and over projective limits of unitary groups, Duke Mathematical Journal, Vol. 114, No. 2, 239–266, (2002).
  • [17] D. Pickrell, Measures on infinite dimensional Grassmann manifolds, Journal of Functional Analysis, Vol. 70, Issue 2, 323–356, (1987).
  • [18] B. Rider, B. Valko, Matrix Dufresne Identities, International Mathematics Research Notices, Vol. 2016, Issue 1, 174–218, (2016).
  • [19] D.W. Stroock, Partial Differential Equations for Probabilists, Cambridge Studies in Advanced Mathematics, Vol. 112, (2008).
  • [20] S. Vakeroudis, Bougerol’s identity in law and extensions, Probability Surveys, Vol. 9, 411–437, (2012).
  • [21] M. Yor, Exponential Functionals of Brownian Motion and Related Processes, Springer Finance, (2001).