Hélène Airault

Modulus of continuity of the canonic Brownian motion “on the group of diffeomorphisms of the circle”

Abstract An explicit modulus of Holder continuity is given for the flow associated to the canonic Brownian motion on the diffeomorphism group of the circle.

An algebra of differential operators and generating functions on the set of univalent functions

Abstract With a method close to that of Kirillov [4], we define sequences of vector fields on the set of univalent functions and we construct systems of partial differential equations which have the sequence of the Faber polynomials ( F n ) as a solution. Through the Faber polynomials and Grunsky coefficients, we obtain the generating functions for some of the sequences of vector fields.

Smoothness of local times of semimartingales

In this paper, applying the K -method in the real interpolation theory we show that for some semimartingales, their local times considered as functionals on the Wiener space belong to the fractional Sobolev spaces on the Wiener space Dpα for p>1 and α<1/2 . Moreover, for the Brownian motion, we can prove that the result for the regularity of the local time as a functional on the Wiener space is optimal.

Support of Virasoro unitarizing measures

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 Mobius transformations. To cite this article: H. Airault et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 621-626.  2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS

On the action of Virasoro algebra on the space of univalent functions

We obtain explicit expressions for differential operators defining the action of the Virasoro algebra on the space of univalent functions. We also obtain an explicit Taylor decomposition for Schwarzian derivative and a formula for the Grunsky coefficients.

Mesure unitarisante : algèbre de Heisenberg, algèbre de Virasoro

Let R2∞ be the infinite product of countably many copies of R2. A Borelian probability measure on the infinite dimensional topological space R2∞ which is unitarizing for the canonical representation of the infinite dimensional Heisenberg algebra is a Gaussian measure on R2∞. To cite this article: H. Airault, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 787–792.

Smoothness of stopping timesof diffusion processes

For an elliptic diffusion process, we prove that the exit time from an open set is in the fractional Sobolev spaces Epα (or Dpα) provided that pα<1. The result is almost optimal.

Smoothness of indicator functions of some sets in Wiener spaces

Abstract Let A be a ball in the classical Wiener space defined by the sup-norm or the pseudo-Sobolev norm, then χ A ∈ E p α for pα χ A ∉ E p α for pα >1.

Affine coordinates and Virasoro unitarizing measures

Abstract Two univalent functions with the same Schwarzian derivative are equivalent. Extending the Kirillov method of identifying a univalent function with the sequence of the coefficients in its asymptotic expansion, a realization of the integration by parts formula for unitarizing measures of the Virasoro algebra is given in affine coordinates.

Functorial Analysis in Geometric Probability Theory

We define the image of a connection with the help of a functor. This image is a connection; we study its properties and also the stability of various Riemannian formulae through this functor.