Annales de la Faculté des Sciences de Toulouse

Yimin Xiao

Properties of local-nondeterminism of Gaussian and stable random fields and their applications

Gilcione Nonato Costa

Holomorphic foliations by curves on $\mathbb{P}^3$ with non-isolated singularities

William Alexandre

$C^k$-estimates for the $\overline{\partial }$-equation on concave domains of finite type

Frédéric Menous

An example of local analytic q-difference equation : Analytic classification

Serge Cohen

Jacques Istas

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
SERGE COHEN & JACQUES ISTAS
Introduction
Tome XV, no 1 (2006), p. 1-2.
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Second conference
on self-similarity and applications
Self-similarity arises from various problems in signal processing, stochas-
tic calculus, turbulence, multifractal phenomena, among others. The aim of
the Second Conference on Self-similarity and Applications, held in Toulouse
from 20 to 24 of June 2005, was to offer a large overview on self-similarity
and its applications. This special issue of the Annales de la Faculté des
Sciences de Toulouse consists of research and expository papers of the in-
vited talks. Let us briefly introduce these papers.
How can be formalized the idea that a function satisfies some scaling
property in the neighbourhood of a given point? This question is seminal
when working in a self-similar setting, and Stéphane Jaffard introduces and
studies a general definition of pointwise regularity associated with a quasi-
Banach space which is shift and scaling invariant.
Since the Cantor set, deterministic or random trees have been used for
modeling fractal sets. Jean-François Le Gall discusses the class of real self-
similar random trees, which generalize the Continuous Random Trees, and
explains their connections with superprocesses.
Stochastic calculus with respect to Brownian Motion is now a stan-
dard topic. Fractional Brownian Motion is the archetypal self-similar process
and stochastic calculus with respect to it is an ongoing expansion theme.
David Nualart provides us with a survey on different approaches intro-
duced recently to construct stochastic integrals with respect to Fractional
Brownian Motion: pathwise techniques, Malliavin calculus, approximation
by Riemann sums.
Multiscale analysis of data is a usual tool in physics. Tools like wavelets
and fractals are currently used and Sylvie Roques and Frédérique Rémi
detail the multiscale analysis of solar cycles.
Long range dependence is a fascinant subject since B. Mandelbrot
brought the attention of the scientific community to the water levels in
the Nile river. Gennady Samorodnitsky gives the main ideas and results on
long memory, scaling and self-similarity.
Many problems of statistical inference appear when working in a stochas-
tic self-similar setting. Some of them are linked to the occupation measure of
– 1 –
Second conference on self-similarity and applications
the path of the stochastic process. Mario Wschebor presents both classical
and recent results on approximation theorems of the occupation measure.
Some questions, like small ball properties, Hausdorff dimensions and
measures, require sharp estimates of sample path properties. Yimin Xiao
shows how these properties should be linked to local nondeterministic self-
similar stochastic processes.
At least, we hope that this issue provides a real survey on topical ques-
tions related to self-similarity.
Serge Cohen and Jacques Istas
– 2 –


Second conference on self-similarity and applications

Sylvie Roques

Frédérique Rémy

Dissection multi-échelle de quelques systèmes naturels : de la cryosphère Antarctique aux pulsations solaires

Jean Chanzy

Inverse du Laplacien discret dans le problème de Poisson-Dirichlet à deux dimensions sur un rectangle

Romain Bondil

Geometry of superficial elements

Luigi Ambrosio

Gianluca Crippa

Stefania Maniglia

Traces and fine properties of a $BD$ class of vector fields and applications

Dominique Bakry

Abdellatif Bentaleb

Extension of Bochner-Lichnérowicz formula on spheres

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Properties of local-nondeterminism of Gaussian and stable random fields and their applications

Holomorphic foliations by curves on $\mathbb{P}^3$ with non-isolated singularities

$C^k$ -estimates for the $\overline{\partial }$ -equation on concave domains of finite type

An example of local analytic q-difference equation : Analytic classification

Second conference on self-similarity and applications

Dissection multi-échelle de quelques systèmes naturels : de la cryosphère Antarctique aux pulsations solaires

Inverse du Laplacien discret dans le problème de Poisson-Dirichlet à deux dimensions sur un rectangle

Geometry of superficial elements

Traces and fine properties of a $BD$ class of vector fields and applications

Extension of Bochner-Lichnérowicz formula on spheres

1,744

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