Many phenomena of interest are analyzed and controlled
through graphs or n-dimensional images. Often, these graphs have
an *irregular aspect*, whether the studied phenomenon is of natural
or artificial origin. In the first class, one may cite
natural landscapes, most biological signals and images (EEG, ECG, MR images, ...),
and temperature records. In the second class, prominent examples include financial logs and TCP traces.

Such irregular phenomena are usually not adequately described by purely deterministic models, and a probabilistic ingredient is often added. Stochastic processes allow to take into account, with a firm theoretical basis, the numerous microscopic fluctuations that shape the phenomenon.

In general, it is a wrong view to believe that
irregularity appears as an epiphenomenon, that is
conveniently dealt with by introducing randomness. In many situations, and
in particular in some of the examples
mentioned above, irregularity is a core
ingredient that cannot be removed without destroying the
phenomenon itself. In some cases, irregularity is even a
necessary condition for proper functioning.
A striking example is that of ECG: an ECG is inherently irregular, and, moreover, in a mathematically precise
sense, an *increase* in its regularity is strongly correlated with a *degradation* of its condition.

In fact, in various situations, irregularity is a crucial feature that can be used
to assess the behaviour of a given system. For instance,
irregularity may the result of two or more sub-systems that
act in a concurrent way to achieve some kind of equilibrium.
Examples of this abound in nature
(*e.g.* the sympathetic and parasympathetic systems in the regulation of the heart). For artifacts, such as financial logs and TCP traffic, irregularity is in a sense
an unwanted feature, since it typically makes regulations more complex. It is
again, however, a necessary one. For instance, efficiency in financial markets requires a constant flow of information among agents, which manifests itself
through permanent fluctuations of the prices: irregularity just reflects the evolution of this information.

The aim of *Regularity* is a to develop a coherent set of methods allowing to model such “essentially
irregular” phenomena in view of managing the uncertainties entailed by their irregularity.

Indeed, essential irregularity makes it more to difficult to study phenomena in terms of their description,
modeling, prediction and control. It introduces *uncertainties* both in
the measurements and the dynamics. It is, for instance, obviously easier to predict the short
time behaviour of a smooth (*e.g.*

The modeling of essentially irregular phenomena is an important challenge, with an emphasis on understanding the sources and functions of this irregularity. Probabilistic tools are well-adapted to this task, provided one can design stochastic models for which the regularity can be measured and controlled precisely. Two points deserve special attention:

first, the study of regularity has to be *local*. Indeed, in most applications, one will want to act on a system based on local temporal or spatial information. For instance, detection of arrhythmias in ECG
or of krachs in financial markets should be performed in “real time”, or, even better, ahead of time. In this sense, regularity is a *local* indicator of the *local* health of a system.

Second, although we have used the term “irregularity” in a generic and somewhat vague sense, it seems obvious that, in real-world phenomena, regularity comes in many colors, and a rigorous analysis should distinguish between them. As an example, at least two kinds of irregularities are present in financial logs: the local “roughness” of the records, and the local density and height of jumps. These correspond to two different concepts of regularity (in technical terms, Hölder exponents and local index of stability), and they both contribute a different manner to financial risk.

In view of the above, the *Regularity* team focuses on the design of methods that:

define and study precisely various relevant measures of local regularity,

allow to build stochastic models versatile enough to mimic the rapid variations of the different kinds of regularities observed in real phenomena,

allow to estimate as precisely and rapidly as possible these regularities, so as to alert systems in charge of control.

Our aim is to address the three items above through the design of mathematical tools in the field of probability (and, to a lesser extent, statistics), and to apply these tools to uncertainty management as described in the following section. We note here that we do not intend to address the problem of controlling the phenomena based on regularity, that would naturally constitute an item 4 in the list above. Indeed, while we strongly believe that generic tools may be designed to measure and model regularity, and that these tools may be used to analyze real-world applications, in particular in the field of uncertainty management, it is clear that, when it comes to control, application-specific tools are required, that we do not wish to address.

The research topics of the *Regularity* team can be roughly divided into two strongly interacting axes, corresponding to two complementary ways of studying regularity:

developments of tools allowing to characterize, measure and estimate various notions of local regularity, with a particular emphasis on the stochastic frame,

definition and fine analysis of stochastic models for which some aspects of local regularity may be prescribed.

These two aspects are detailed in sections and below.

**Fractional Dimensions**

Although the main focus of our team is on characterizing *local*
regularity, on occasions, it is interesting to use a *global*
index of regularity. Fractional dimensions provide such an index.
In particular, the *regularization dimension*, that was defined
in , is well adapted to the study stochastic processes, as its
definition allows to build robust estimators in an easy way.
Since its introduction, regularization dimension has been used by various teams
worldwide in many different applications including the characterization of certain stochastic
processes, statistical estimation,
the study of mammographies or galactograms for breast
carcinomas detection,
ECG analysis for the study of ventricular arrhythmia,
encephalitis diagnosis from EEG, human skin analysis,
discrimination between the nature of radioactive contaminations,
analysis of porous media textures,
well-logs data analysis,
agro-alimentary image analysis, road profile analysis, remote sensing,
mechanical systems assessment, analysis of video games, ...(see http://

**Hölder exponents**

The simplest and most popular measures of local
regularity are the pointwise
and local Hölder exponents. For a stochastic process

and

Although these quantities are in general random, we will omit as is customary
the dependency in

The random functions

The pointwise Hölder exponent is a very versatile
tool, in the sense that the set of pointwise Hölder functions of
continuous functions is quite large (it coincides with the set of
lower limits of sequences of continuous functions ). In this sense,
the pointwise exponent is often a more precise tool
(*i.e.* it varies in a more rapid way)
than the local one, since local Hölder functions are always lower semi-continuous.
This is why, in particular, it is
the exponent that is used as a basis ingredient in multifractal
analysis (see section ). For certain classes of stochastic
processes, and most notably Gaussian processes, it has the remarkable
property that, at each point, it assumes an almost sure value .
SRP, mBm, and processes of this kind (see sections and
) rely on the sole use
of the pointwise Hölder exponent for prescribing the regularity.

However,

Another, related, drawback of the pointwise exponent is that it is
not stable under integro-differentiation, which sometimes makes
its use complicated in applications. Again, the local exponent provides
here a useful complement to

Both exponents have proved useful in various applications, ranging from image denoising and segmentation to TCP traffic characterization. Applications require precise estimation of these exponents.

**Stochastic 2-microlocal analysis**

Neither the pointwise nor the local exponents give a complete characterization of the local regularity, and, although their joint use somewhat improves the situation, it is far from yielding the complete picture.

A fuller description of local regularity is provided by the
so-called *2-microlocal analysis*, introduced by J.M. Bony
. In this frame, regularity
at each point is now specified by two indices, which makes the analysis
and estimation tasks more difficult. More precisely,
a function *2-microlocal space*

for all *2-microlocal
spectrum*. This spectrum provide a wealth of information on the local
regularity.

In , we have laid some foundations for a stochastic version of 2-microlocal analysis. We believe this will provide a fine analysis of the local regularity of random processes in a direction different from the one detailed for instance in .We have defined random versions of the 2-microlocal spaces, and given almost sure conditions for continuous processes to belong to such spaces. More precise results have also been obtained for Gaussian processes. A preliminary investigation of the 2-microlocal behaviour of Wiener integrals has been performed.

**Multifractal analysis of stochastic processes**

A direct use of the local regularity is often fruitful in applications.
This is for instance the case in RR analysis or terrain
modeling. However, in some situations,
it is interesting to supplement or replace it by a more global
approach known as *multifractal analysis* (MA). The idea behind
MA is to group together all points with same regularity (as measured
by the pointwise Hölder exponent) and to measure the “size” of
the sets thus obtained , , . There are mainly two ways to do so, a geometrical
and a statistical one.

In the geometrical approach, one defines the
*Hausdorff multifractal spectrum* of a process or function

The statistical path to MA is based on the so-called
*large deviation multifractal spectrum*:

where:

and *i.e.*:

Here,

The large deviation spectrum is typically easier to compute and to estimate than the Hausdorff one. In addition, it often gives more relevant information in applications.

Under very mild conditions (*e.g.* for instance, if
the support of *Legendre multifractal spectrum*. To do so,
one basically interprets the spectrum

with the convention

The Legendre multifractal spectrum of

To see the relation between

where *weak multifractal
formalism* holds, *i.e.* *strong
multifractal formalism*.

Multifractal spectra subsume a lot of information about the distribution of the regularity, that has proved useful in various situations. A most notable example is the strong correlation reported recently in several works between the narrowing of the multifractal spectrum of ECG and certain pathologies of the heart , . Let us also mention the multifractality of TCP traffic, that has been both observed experimentally and proved on simplified models of TCP , .

**Another colour in local regularity: jumps**

As noted above, apart from Hölder exponents and their generalizations,
at least another type of irregularity may sometimes be observed on
certain real phenomena: discontinuities, which occur for instance
on financial logs and certain biomedical signals. In this frame, it is of
interest to supplement Hölder exponents and their extensions with (at least) an additional
index that measures the local intensity and size of jumps. This is a topic we
intend to pursue in full generality in the near future. So far, we have developed an approach
in the particular frame of *multistable processes*. We refer to section
for more details.

The second axis in the theoretical developments of the *Regularity* team aims at defining and studying stochastic processes for which various aspects of the local regularity may be prescribed.

**Multifractional Brownian motion**

One of the simplest stochastic process for which some kind of control over the Hölder exponents is possible is probably fractional Brownian motion (fBm). This process was defined by Kolmogorov and further studied by Mandelbrot and Van Ness, followed by many authors. The so-called “moving average” definition of fBm reads as follows:

where

Although varying

It is possible to generalize fBm to obtain a Gaussian process for which the pointwise Hölder exponent
may be tuned at each point: the *multifractional Brownian motion (mBm)* is such
an extension, obtained by substituting the constant parameter *regularity function*

mBm was introduced independently by two groups of authors:
on the one hand, Peltier and Levy-Vehel defined the mBm

On the other hand, Benassi, Jaffard and Roux defined the mBm from the harmonizable representation of the
fBm, *i.e.*:

where

The Hölder exponents of the mBm are prescribed almost surely:
the pointwise Hölder exponent is

The fact that the local regularity of mBm
may be tuned *via* a functional parameter has made it a useful
model in various areas such as finance, biomedicine,
geophysics, image analysis, ....
A large number of studies have been devoted worldwide to its mathematical properties,
including in particular its local time. In addition,
there is now a rather strong body of work dealing the estimation of its
functional parameter, *i.e.* its local regularity. See http://

**Self-regulating processes**

We have recently introduced another class of stochastic models, inspired by mBm,
but where the local regularity, instead of being tuned “exogenously”, is
a function of the amplitude. In other words, at each point *self-regulating process* (SRP).
The particular process obtained by adapting adequately mBm is called
the self-regulating multifractional process . Another instance is given by
modifying the Lévy construction of Brownian motion .
The motivation for introducing self-regulating processes is based on the following general fact: in nature, the local regularity of a phenomenon is often related to its amplitude.
An intuitive example is provided by natural terrains: in young mountains, regions
at higher altitudes are typically more irregular than regions at lower altitudes.
We have verified this fact experimentally on several digital elevation models
. Other natural phenomena displaying a relation between
amplitude and exponent include temperatures
records and RR intervals extracted from ECG .

To build the SRMP, one starts from a field of fractional Brownian motions

the affine rescaling between

where

An example of a two dimensional SRMP with function

We believe that SRP open a whole new and very promising area of research.

**Multistable processes**

Non-continuous phenomena are commonly encountered in real-world
applications, *e.g.* financial records or EEG traces.
For such processes, the information brought
by the Hölder exponent must be supplemented by some measure of
the density and size of jumps. Stochastic processes with jumps,
and in particular Lévy processes, are currently an active area of research.

The simplest class of non-continuous Lévy processes is maybe the one
of stable processes . These are mainly characterized by a parameter
*stability index* (

In line with our quest for the characterization and modeling of
various notions of local regularity, we have defined *multistable processes*.
These are processes which are
“locally” stable, but where
the stability index

More formally, a multistable process is a process which is,
at each time

where the limit is understood either in finite dimensional
distributions or in the stronger sense of distributions.
Note

One approach to defining multistable processes is similar to the one
developed for constructing mBm : we consider fields of stochastic processes

A particular class of multistable processes, termed
“linear multistable multifractional
motions” (lmmm) takes the following form , .
Let

where

In fact, lmmm are somewhat more general than said above:
indeed, the couple

Our theoretical works are motivated by and find natural applications to real-world problems in a general frame generally referred to as uncertainty management, that we describe now.

Since a few decades, modeling has gained an increasing part in complex systems design in various fields of industry such as automobile, aeronautics, energy, etc. Industrial design involves several levels of modeling: from behavioural models in preliminary design to finite-elements models aiming at representing sharply physical phenomena. Nowadays, the fundamental challenge of numerical simulation is in designing physical systems while saving the experimentation steps.

As an example, at the early stage of conception in aeronautics, numerical simulation aims at exploring the design parameters space and setting the global variables such that target performances are satisfied. This iterative procedure needs fast multiphysical models. These simplified models are usually calibrated using high-fidelity models or experiments. At each of these levels, modeling requires control of uncertainties due to simplifications of models, numerical errors, data imprecisions, variability of surrounding conditions, etc.

One dilemma in the design by numerical simulation is that many crucial choices are made very early, and thus when uncertainties are maximum, and that these choices have a fundamental impact on the final performances.

Classically, coping with this variability is achieved through *model registration* by experimenting and adding fixed *margins* to the model response.
In view of technical and economical performance, it appears judicious to replace these fixed margins by a rigorous analysis and control of risk. This may be achieved through a probabilistic approach to uncertainties, that provides decision criteria adapted to the management
of unpredictability inherent to design issues.

From the particular case of aircraft design emerge several general aspects of management of uncertainties in simulation. Probabilistic decision criteria, that translate decision making into mathematical/probabilistic terms, require the following three steps to be considered :

build a probabilistic description of the fluctuations of the model's parameters (*Quantification* of uncertainty sources),

deduce the implication of these distribution laws on the model's response (*Propagation* of uncertainties),

and determine the specific influence of each uncertainty source on the model's response variability (*Sensitivity Analysis*).

The previous analysis now constitutes the framework of a general study of uncertainties. It is used in industrial contexts where uncertainties can be represented by *random variables* (unknown temperature of an external surface, physical quantities of a given material, ... at a given *fixed time*). However, in order for the numerical models to describe with high fidelity a phenomenon, the relevant uncertainties must generally depend on time or space variables.
Consequently, one has to tackle the following issues:

*How to capture the distribution law of time (or space) dependent parameters,
without directly accessible data?*
The distribution of probability of the continuous time (or space) uncertainty sources must describe the links between variations at neighbor times (or points).
The local and global regularity are important parameters of these laws, since it describes how the fluctuations at some time (or point) induce fluctuations at close times (or points).
The continuous equations representing the studied phenomena should help *to propose models for the law of the random fields*.
Let us notice that interactions between various levels of modeling might also be used to derive distributions of probability at the lowest one.

The navigation between the various natures of models needs a kind of *metric* which could *mathematically describe the notion of granularity or fineness* of the models.
Of course, the local regularity will not be totally absent of this mathematical definition.

All the various levels of conception, preliminary design or high-fidelity modelling, require *registrations by experimentation* to reduce model errors.
This *calibration* issue has been present in this frame since a long time, especially in a deterministic optimization context. The random modeling of uncertainty requires the definition of a systematic approach.
The difficulty in this specific context is: statistical estimation with few data and estimation of a function with continuous variables using only discrete setting of values.

Moreover, a multi-physical context must be added to these questions. The complex system design is most often located at the interface between several disciplines. In that case, modeling relies on a coupling between several models for the various phenomena and design becomes a *multidisciplinary optimization* problem. In this uncertainty context, the real challenge turns robust optimization to manage technical and economical risks (risk for non-satisfaction of technical specifications, cost control).

We participate in the uncertainties community through several collaborative research projects. As explained above, we focus on essentially irregular phenomena, for which irregularity is a relevant quantity to capture the variability (e.g. certain biomedical signals, terrain modeling, ﬁnancial data, etc.). These will be modeled through stochastic processes with prescribed regularity.

A striking feature of many financial logs is that they are both irregular
in the Hölder sense and
display jumps. Furthermore, the local roughness as well as the size of jumps
typically vary in time. This hints that multifractional multistable processes
may provide well-adapted models. As a first step,
we shall investigate the simple case of multistable Lévy motions and
concentrate on understanding how a time-varying

In another direction, we will study whether multifractional Brownian motion (mBm) and SRP provide useful models in the frame of financial modeling. Fractional Brownian motion-based option pricing and portfolio selection has attracted a lot of interest in recent years. This process is certainly a more adequate model than pure Brownian motion, as many studies have shown. However, it is also clear that it suffers various limitations. One of the most obvious is that the local regularity of financial logs is not constant, as is apparent on any sufficiently long sample. The most direct way of generalizing fractional Brownian motion to account for this fact is to consider mBm, as we have done in , using the theory of stochastic calculus with respect to mBm that we have recently developed in , . Another possibility is to use SRP. This requires to extend both the theoretical results (mainly those related to stochastic calculus) and their applications (pricing, portfolio selection) beyond the case of fractional Brownian motion. A disadvantage of mBm is that, in order to price for instance, one has to know the regularity function ahead of time, which usually requires additional assumptions, or to build a model for its evolution. This problem is not present for the SRP: no further information is required once the function relating the amplitude and the regularity has been identified. On the other hand, stochastic integration with respect to SRP (which is neither a Gaussian process nor a semi-martingale) does not seem to be within reach at present, since little is known indeed about this process. This nevertheless constitutes one of our long term goals.

FracLab was developed for two main purposes:

propose a general platform allowing research teams to avoid the need to re-code basic and advanced techniques in the processing of signals based on (local) regularity.

provide state of the art algorithms allowing both to disseminate new methods in this area and to compare results on a common basis.

FracLab is a general purpose signal and image processing toolbox based on fractal, multifractal and local regularity methods. FracLab can be approached from two different perspectives:

(multi-) fractal and local regularity analysis: A large number of procedures allow to compute various quantities associated with 1D or 2D signals, such as dimensions, Hölder and 2-microlocal exponents or multifractal spectra.

Signal/Image processing: Alternatively, one can use FracLab directly to perform many basic tasks in signal processing, including estimation, detection, denoising, modeling, segmentation, classification, and synthesis.

A graphical interface makes FracLab easy to use and intuitive. In addition, various wavelet-related tools are available in FracLab.

FracLab is a free software. It mainly consists of routines
developed in MatLab or C-code interfaced with MatLab.
It runs under Linux, MacOS and Windows environments. In addition,
a “stand-alone” version (*i.e.* which does not require
MatLab to run) is available.

Fraclab has been downloaded several thousands of times in the last years
by users all around the world. A few dozens
laboratories seem to use it regularly, with more than four hundreds registered users.
Our ambition is to make it the
standard in fractal softwares for signal and image processing
applications. We have signs that this is starting to become
the case. To date, its use has been acknowledged in roughly three hundreds and fifty
research papers in various areas such as astrophysics, chemical engineering,
financial modeling, fluid dynamics, internet and road traffic analysis, image and signal processing,
geophysics, biomedical applications, computer science, as well as in mathematical studies in analysis and
statistics (see http://

The article "Christiane's Hair" by Jacques Lévy-Véhel and Franklin Mendivil has received the Paul R. Halmos - Lester R. Ford award of the Mathematical Association of America.

*In collaboration with Pierre Emmanuel Lévy Véhel and Victor Lévy Véhel.*

Illegal sharing of cultural goods on the Internet has become a massive reality in today's connected society. Numerous studies have been performed to try and evaluate the impact of these practices on the industry of cultural goods, and how much harm, if any, they have entailed. The effect of legal and technical responses to limit pirating has also been investigated, showing in general inconclusive effect. Instead of penalizing illegal actors - providers and/or consumers -, a totally different approach has been proposed recently by the french government agency Hadopi. The idea is to offer the possibility to sites that illegally share cultural goods to become legal in exchange of a retribution proportional to their activity. In the frame of a contract with the Hadopi, we have built a model that studies the economic feasibility of such a scheme under various assumptions on the behaviour of the different actors involved. Our main finding is that, supposing that more popular goods are more prone to pirating, a retribution of the order of the increase in benefit per user gained by legalized sites does indeed lead to a win-win situation for both producers/sellers of cultural goods and willing-to-be-legalized sites. This will be the case under two conditions: the proportion of pirates is large enough (which seems largely true) and the increase in the amount of money that forums will make from advertisement when becoming legal is sufficient .

An extension of our work is under way, that will consider further actors and refined modelling of the way illegal sharing takes place. Calibration issues will also be investigated more closely.

Financial regulations have fundamentally changed since the Basel II Accords. Among other evolutions, Basel II and III explicitly impose that computations of capital requirements be model-based. This paradigm shift in risk management has been the source of strong debates among both practitioners and academics, who question whether such model-based regulations are indeed more efficient.

A common feeling in the industry is that regulations will sometimes give a false impression of security: risk manager tend to think that a financial company that would fulfil all the criteria of, say, the Basel III Accords on capital adequacy, is not necessarily on the safe side. This is so mainly because many risks, and most significantly systemic or system-wide risks, are not properly modelled, and also because it is easy to manipulate to some extent various risk measures, such as VaR.

In parallel, a fast growing body of academic research provides various arguments explaining why current regulations are not well fitted to address risk management in an adequate way, and may even, in certain cases, worsen the situation.

We use the term *regulation risk* to describe the fact that,
in some situations, prudential rules are
themselves the source of a systemic risk. We have shown how a combination
of model risk and regulation risk leads to an effect which is exactly
the opposite of what the regulator tries to enforce. More precisely, we
explain how wrongly assuming a Gaussian dynamics (or, more generally, a
left-light-tailed one) when the “true” one is pure jump (or, more generally,
left-heavy-tailed), and imposing as a constraint *minimizing* VaR at constant
volume results in effect in movements that will *maximize* VaR.
This effect is related to the fact that regulations fail to consider
that risk is endogenous.
In a nutshell, the idea is simply that, by treating jumps
in the evolution of prices as exceptional events and essentially ignoring them
in model-based VaR computations, one misses an essential dimension of risk,
and acts in a way that will in effect favour sudden large movements in the
markets and ultimately increase VaR. Our simple setting predicts that VaR
constraints result in an *increased* intensity of jumps and a
*decrease* in volatility - a fact confirmed
experimentally on certain datasets. This is a mathematical translation
of the common feeling of practitioners that regulations give a false impression of
security characterized by low volatility but increased risk of sudden large movements.

We prove a functional central limit theorem (FCLT) for the independent-increments multistable Lévy motions (MsLM)

**Theorem 0.1** Let

tends in distribution to

uniformly for all

We have defined integrals of MsLM, and given criteria for convergence,independence, stochastic Hölder continuity and strong localisability of such integrals.

In the papers , we study some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett (1962), Freedman (1975), van de Geer (1995), de la Peña (1999) and Pinelis (2006). Moreover, our concentration inequalities also improve some known inequalities for sums of independent random variables. Applications associated with linear regressions, autoregressive processes and branching processes are provided. In particular, an interesting application of de la Peña's inequality to self-normalized deviations is also provided.

We also considered an

Self-stabilizing processes have the property that the “local intensities of jumps” varies with amplitude. They are good models for, e.g., financial and temperature records.

The main aim of our work is to establish the existence of such processes and to give a simple construction.
Formally, one says that a stochastic process

where convergence is in finite dimensional distributions with respect to

when

This inspiration allows us to build Markov processes that converge to a self-stabilizing process. Note that, when

**Definition 0.1** We call the sequence

whenever

The slightly generalized version of the Arzelà-Ascoli theorem reads:

**Lemma 0.1** Assume that

The following theorem states that self-stabilizing processes do exist.

**Theorem 0.2** Let

We are currently studying the main properties of self-stabilizing processes.

The Tandem Project is a consortium involving several industrial companies (e.g. Bull Amesys) and some research laboratories (e.g. CMAP). The aim is to detect landmines from 3D radar images.

Hadopi contract on the economical feasibility of a way to reduce pirating of cultural goods on the Internet.

Regularity has strong collaborations with Nantes University (Anne Philippe) and Rennes University (Ronan Le Guével) .

Regularity collaborates with St Andrews University (Prof. Kenneth Falconer) on the study of multistable processes.

Regularity collaborates with Acadia University (Prof. Franklin Mendivil) on the study of fractal strings, certain fractals sets, and the study of the regularization dimension.

Pr. Franklin Mendivil, from Acadia University was invited for one month in the team.

Regularity has organized and hosted a conference in honour of Pr. K. Falconer's 60th birthday in May 2014.

Jacques Lévy Véhel is associate editor of the journal *Fractals*.

Xiequan Fan is a reviewer for Mathematical Reviews (AMS). Jacques Lévy Véhel reviewed papers for many journals and conferences.

Master: Jacques Lévy Véhel, Wavelets and Fractals, M2, 8h, Ecole Centrale Nantes.

Master: Jacques Lévy Véhel, Wavelets and Fractals, M2, 18h, ESIEA.

PhD : Benjamin Arras, Around some selfsimilar processes with stationary increments, Ecole Centrale Paris, December 2014, advisor : J. Lévy Véhel

PhD : Alexandre Richard, Local regularity of some fractional Brownian ﬁelds, Ecole Centrale Paris, September 2014, advisor : E. Merzbach

J. Lévy Véhel has been a member of the juries for recruiting two AER, one AS and one AF at Inria Saclay.

J. Lévy Véhel is a member of the Bureau du Comité des Projets, of the Commission Scientifique, and of the Comité de Centre at Inria Saclay. He is the animator of the Commission de Suivi Doctoral also at Inria Saclay. Finally, he was the head of the jury for the 2014 CR2 positions contest for Inria Saclay.