Realization of Virasoro unitarizing measures on the set of Jordan curves

Helene Airault; Vladimir Bogachev

doi:10.1016/S1631-073X(03)00085-2

Probability Theory

Realization of Virasoro unitarizing measures on the set of Jordan curves
[Réalisation de mesures unitarisantes sur l'ensemble des courbes de Jordan]

Présenté par : Paul Malliavin

Helene Airault ^1,² ; Vladimir Bogachev ³

¹ INSSET, Université de Picardie, 48, rue Raspail, 02100 Saint-Quentin (Aisne), France
² Laboratoire CNRS UMR 6140, LAMFA, 33, rue Saint-Leu, 80039 Amiens, France
³ Dept. Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia

Comptes Rendus. Mathématique, Volume 336 (2003) no. 5, pp. 429-434.

Résumé
Abstract

Deux fonctions univalentes sont équivalentes, f∼g, si elles ont même dérivée Schwarzienne. La relation d'équivalence ∼ étant définie à une transformation homographique près, on obtient un isomorphisme entre la variété $𝒥$ des courbes de Jordan et la variété quotient $\tilde{𝒮}$ . Cela permet de déduire des champs de vecteurs sur $\tilde{𝒮}$ et sur $𝒥$ . On explicite l'action de ces champs de vecteurs sur les polynômes de Neretin. On étudie l'existence de mesures unitarisantes sur le quotient de l'ensemble des fonctions univalentes par cette relation d'équivalence et pour une telle mesure, on établit des relations d'orthogonalité entre les polynômes de Neretin. Ce travail est une réalisation concrète du quotient $Diff (S^{1}) / SL (2, R)$ de Airault et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 621–626 sur l'espace complexe $C^{\infty}$ produit d'une infinité dénombrable de $C$ .

Two univalent functions are equivalent, f∼g, if they have the same Schwarzian derivative. The equivalence relation ∼ being defined up to an homographic transformation, it gives an isomorphism between the manifold $𝒥$ of Jordan curves and the quotient manifold $\tilde{𝒮}$ . It permits to obtain vector fields on $\tilde{𝒮}$ and on $𝒥$ . The action of these vector fields on the Neretin polynomials is explicited. The existence of a unitarizing measure on the quotient manifold $\tilde{𝒮}$ is discussed and for such a measure, orthogonality relations for the Neretin polynomials are obtained. This work is a concrete realization on the complex space $C^{\infty}$ of the abstract quotient $Diff (S^{1}) / SL (2, R)$ considered in Airault et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 621–626.

Reçu le : 2002-12-18
Accepté le : 2003-01-08
Publié le : 2003-03-01

DOI : 10.1016/S1631-073X(03)00085-2

Affiliations des auteurs :

Helene Airault ^{1, 2} ; Vladimir Bogachev ³

¹ INSSET, Université de Picardie, 48, rue Raspail, 02100 Saint-Quentin (Aisne), France
² Laboratoire CNRS UMR 6140, LAMFA, 33, rue Saint-Leu, 80039 Amiens, France
³ Dept. Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia

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Helene Airault; Vladimir Bogachev. Realization of Virasoro unitarizing measures on the set of Jordan curves. Comptes Rendus. Mathématique, Volume 336 (2003) no. 5, pp. 429-434. doi : 10.1016/S1631-073X(03)00085-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00085-2/

Bibliographie
Cité par

[1] H. Airault Mesure unitarisante; algèbre de Heisenberg, algèbre de Virasoro, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002), pp. 787-792

[2] H. Airault; P. Malliavin Unitarizing probability measures for representations of Virasoro algebra, J. Math. Pures Appl., Volume 80 (2001) no. 6, pp. 627-667

[3] H. Airault; P. Malliavin; A. Thalmaier Support of Virasoro unitarizing measures, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 621-626

[4] H. Airault; J. Ren An algebra of differential operators and generating functions on the set of univalent functions, Bull. Sci. Math. (2002)

[5] V. Bogachev Cours à l'Université de Pise. Differentiable measures and the Malliavin calculus (Mai 1995), Scuola Normale Superiore, Pisa, J. Math. Sci., Volume 87 (1997) no. 4, pp. 3577-3731

[6] A.A. Kirillov Geometric approach to discrete series of unireps for Virasoro, J. Math. Pures Appl., Volume 77 (1998), pp. 735-746

[7] O. Lehto Univalent Functions and Teichmüller Spaces, Graduate Texts in Math., 109, Springer-Verlag, 1987

[8] Yu.A. Neretin Holomorphic extensions of representations of the group of diffeomorphisms of the circle, Math. USSR-Sb., Volume 67 (1990) no. 1, pp. 75-96

Maria Gordina; Wei Qian; Yilin Wang Infinitesimal conformal restriction and unitarizing measures for Virasoro algebra, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 195 (2025), p. 24 (Id/No 103669) | DOI:10.1016/j.matpur.2025.103669 | Zbl:7991277
Hélène Airault; Paul Malliavin; Anton Thalmaier Brownian measures on Jordan-Virasoro curves associated to the Weil-Petersson metric, Journal of Functional Analysis, Volume 259 (2010) no. 12, pp. 3037-3079 | DOI:10.1016/j.jfa.2010.08.002 | Zbl:1213.60129
Hélène Airault From Diff (S 1) to Univalent Functions. Cases of Degeneracy, Analysis and Mathematical Physics (2009), p. 1 | DOI:10.1007/978-3-7643-9906-1_1
Paul Lescot Unitarizing measures for a representation of the Virasoro algebra, according to Kirillov and Malliavin: state of the problem, Mathematical analysis of random phenomena. Proceedings of the international conference, Hammamet, Tunisia, September 12–17, 2005, Hackensack, NJ: World Scientific, 2007, pp. 141-153 | Zbl:1129.58014
M. Gordina; P. Lescot Riemannian geometry of $Diff (S^{1}) / S^{1}$ , Journal of Functional Analysis, Volume 239 (2006) no. 2, pp. 611-630 | DOI:10.1016/j.jfa.2006.02.005 | Zbl:1109.58012
Hélène Airault; Paul Malliavin Quasi-invariance of Brownian measures on the group of circle homeomorphisms and infinite-dimensional Riemannian geometry, Journal of Functional Analysis, Volume 241 (2006) no. 1, pp. 99-142 | DOI:10.1016/j.jfa.2006.01.015 | Zbl:1111.58032
Paul Malliavin Heat Measures and Unitarizing Measures for Berezinian Representations on the Space of Univalent Functions in the Unit Disk, Perspectives in Analysis, Volume 27 (2005), p. 253 | DOI:10.1007/3-540-30434-7_11
Hélène Airault; Paul Malliavin; Anton Thalmaier Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows, Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 83 (2004) no. 8, pp. 955-1018 | DOI:10.1016/j.matpur.2004.06.001 | Zbl:1061.60085
Hélène Airault Affine coordinates and Virasoro unitarizing measures., Journal de Mathématiques Pures et Appliquées. Neuvième Série, Volume 82 (2003) no. 4, pp. 425-455 | DOI:10.1016/s0021-7824(03)00028-x | Zbl:1062.30020

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