Canonical Brownian Motion on the Diffeomorphism Group of the Circle

Fang, Shizan

doi:10.1006/jfan.2002.3922

Cited by 47 publications

(49 citation statements)

References 10 publications

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“…It results from Kunita's theory of stochastic flows that θ → g r x,t (θ ) is a C ∞ diffeomorphism. The limit lim r→1 g r x,t = g x,t exists uniformly in θ and defines a random homeomorphism g x,t , called canonical Brownian motion on Diff(S 1 ), see [23,5,9,26]. This random homeomorphism is furthermore Hölder continuous, see [5,9].…”

Section: Theorem 11 There Exists Up To Multiplication By a Constanmentioning

confidence: 99%

Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows

Airault

Malliavin²,

Thalmaier

2004

Journal de Mathématiques Pures et Appliqués

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A.A. Kirillov has given a parametrization of the space U ∞ of univalent functions on the closed unit disk, which are C ∞ up to the boundary, by Diff(S 1 )/S 1 where Diff(S 1 ) denotes the group of orientation preserving diffeomorphisms of the circle S 1 . In the same spirit, the space J ∞ of C ∞ Jordan curves in the complex plane can be parametrized by the double quotient SU(1, 1)\ Diff(S 1 )/ SU(1, 1). As a consequence, J ∞ carries a canonical Riemannian metric. We construct a canonical Brownian motion on U ∞ . Appl. 83 (2004) Résumé A.A. Kirillov a paramétré l'espace U ∞ des fonctions C ∞ , bijectives dans le disque unité fermé et holomorphes dans son intérieur, par l'espace Diff(S 1 )/S 1 , où Diff(S 1 ) désigne le groupe des difféo-morphismes préservant l'orientation du cercle unité. Dans le même esprit, l'espace J ∞ des courbes de Jordan C ∞ du plan complexe peut être paramétré par SU(1, 1)\ Diff(S 1 )/ SU(1, 1). Par voie de conséquence, l'espace J ∞ possède une métrique riemannienne canonique. Nous construisons dans cet article le mouvement brownien canonique sur U ∞ à l'aide de technologies classiques de la théorie des fonctions univalentes, comme l'extension de Beurling-Ahlfors, l'équation de Loewner, l'équation de Beltrami, revisitées dans le contexte des flots stochastiques à la Kunita. Cet article nous semble être un premier pas vers la construction du mouvement brownien canonique sur J ∞ .  2004 Elsevier SAS. All rights reserved.

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Section: Theorem 11 There Exists Up To Multiplication By a Constanmentioning

confidence: 99%

Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows

Airault

Malliavin²,

Thalmaier

2004

Journal de Mathématiques Pures et Appliqués

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“…Theorem 10 (Theorem of integration by part ( [7,9,11])). Let v ∈ G ρ ; assume that there exists δ > 0 such that, in the sense of the order of hermitian operator on G ρ , the following inequality holds true ρ Ricci ≥ −δ×Identity, then…”

Section: Integration By Part For the Regularized Metricmentioning

confidence: 99%

Itô Atlas, its Application to Mathematical Finance and to Exponentiation of Infinite Dimensional Lie Algebras

Malliavin¹

2007

Stochastic Analysis and Applications

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“…Dans [6,5,3] We denote Diff(S 1 ) the group of C ∞ orientation preserving diffeomorphisms of the circle. The Kähler form used in [2] on M := Diff(S 1 )/S 1 is not positive definite, see [4].…”

Section: Version Française Abrégéementioning

confidence: 99%

“…-The statement is equivalent to [5] for the stochastic process associated to this horizontal Laplacian. To the Kähler metric on M 1 the following Laplacian is associated:…”

Section: Horizontal Laplacian Above Mmentioning

confidence: 99%

Support of Virasoro unitarizing measures

Airault

Malliavin²,

Thalmaier

2002

Comptes Rendus. Mathématique

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A unitarizing measure is a probability measure such that the associated L 2 space contains a closed subspace of holomorphic functionals on which the Virasoro algebra acts unitarily. It has been shown that the unitarizing property is equivalent to an a priori given formula of integration by parts, which has been computed explicitly. We show in this Note that unitarizing measures must be supported by the quotient of the homeomorphism group of the circle by the subgroup of Möbius transformations.

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Canonical Brownian Motion on the Diffeomorphism Group of the Circle

Cited by 47 publications

References 10 publications

Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows

Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows

Itô Atlas, its Application to Mathematical Finance and to Exponentiation of Infinite Dimensional Lie Algebras

Support of Virasoro unitarizing measures

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