Regularity is a comon team between Inria and Ecole Centrale Paris. It is located in the MAS laboratory at Ecole Centrale Paris.

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.*
C^{1}) process than of a nowhere differentiable one.
Likewise, sampling rough functions yields less precise
information than regular ones. As a consequence, when dealing
with essentially irregular phenomena, uncertainties are
fundamental in the sense that one cannot hope to remove them
by a more careful analysis or a more adequate modeling. The
study of such phenomena then requires to develop specific
approaches allowing to manage in an efficient way these
inherent uncertainties.

The paper
*The Estimation of Hölderian Regularity using Genetic
Programming*by Leonardo Trujillo, Pierrick Legrand and
Jacques Levy Vehel won the Best Paper Award for the GP track
in the conference Gecco 2010.

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
whose trajectories are continuous and nowhere
differentiable, these are defined, at a point
t_{0}, as the random variables:

and

Although these quantities are in general
random, we will omit as is customary the dependency in
and
Xand write
(
t_{0})and
instead of
_{X}(
t_{0},
)and
.

The random functions
and
are called respectively the pointwise and local Hölder
functions of the process
X.

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,
_{X}obviously does not give a complete description of
local regularity, even for continuous processes. It is for
instance insensitive to “oscillations”, contrarily to the
local exponent. A simple example in the deterministic frame
is provided by the function
, where
,
are
positive real numbers. This so-called “chirp function”
exhibits two kinds of irregularities: the first one, due to
the term
is measured by the pointwise Hölder exponent. Indeed,
(0) =
.
The second one is due to the wild oscillations around 0, to
which
is blind. In contrast, the local Hölder exponent at 0
is equal to
, and is thus influenced by the oscillatory
behaviour.

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 , since is stable under integro-differentiation.

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
fis said to belong to the
*2-microlocal space*
C_{x0}^{s,
s'}, where
s+
s^{'}>0,
s^{'}<0, if and only if its
m= [
s+
s^{'}]-th order derivative exists around
x_{0}, and if there exists
>0, a polynomial
Pwith degree lower than
[
s]-
m, and a constant
C, such that

for all
x,
ysuch that
0<|
x-
x
_{0}|<
,
0<|
y-
x
_{0}|<
.
This characterization was obtained in
,
. See
,
for other characterizations and
results. These spaces are stable through
integro-differentiation, i.e.
fC_{x}^{s,
s'}if and only if
f^{'}C_{x}^{s-1,
s'}. Knowing to which space
fbelongs thus allows to predict the evolution of its
regularity after derivation, a useful feature if one uses
models based on some kind differential equations. A lot of
work remains to be done in this area, in order to obtain more
general characterizations, to develop robust estimation
methods, and to extend the “2-microlocal formalism” : this is
a tool allowing to detect which space a function belongs to,
from the computation of the Legendre transform of an
auxiliary function known as its
*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
Xas the function:
, where
dim
Edenotes the Hausdorff dimension of
the set
E. This gives a fine measure-theoretic information, but
is often difficult to compute theoretically, and almost
impossible to estimate on numerical data.

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

where:

and
_{n}^{k}is the “coarse grained exponent” corresponding to the
interval
,
*i.e.*:

Here,
Y_{n}^{k}is some quantity that measures the variation of
Xin the interval
I_{n}^{k}, such as the increment, the oscillation or a wavelet
coefficient.

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
f_{g}is bounded,
) the concave envelope of
f_{g}can be computed easily from an auxiliary function,
called the
*Legendre multifractal spectrum*. To do so, one
basically interprets the spectrum
f_{g}as a rate function in a large deviation principle
(LDP): define, for
,

with the convention
0
^{q}: = 0for all
. Let:

The Legendre multifractal spectrum of
Xis defined as the Legendre transform
^{*}of
:

To see the relation between
f_{g}and
f_{l}, define the sequence of random variables
Z_{n}: = log|
Y_{n}^{k}|where the randomness is through a choice of
kuniformly in
{0, ...,
n-1}. Consider the corresponding
moment generating functions:

where
E_{n}denotes expectation with respect to
P_{n}, the uniform distribution on
{0, ...,
n-1}. A version of Gärtner-Ellis
theorem ensures that if
lim
c
_{n}(
q)exists (in which case it equals
1 +
(
q)), and is differentiable, then
c^{*}=
f_{g}-1. In this case, one says that the
*weak multifractal formalism*holds,
*i.e.*
f_{g}=
f_{l}. In favorable cases, this also coincides with
f_{h}, a situation referred to as the
*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
denotes the real white noise. The parameter
Hranges in
(0, 1), and it governs the
pointwise regularity: indeed, almost surely, at each point,
both the local and pointwise Hölder exponents are equal to
H.

Although varying
Hyields processes with different regularity, the fact
that the exponents are constant along any single path is
often a major drawback for the modeling of real world
phenomena. For instance, fBm has often been used for the
synthesis natural terrains. This is not satisfactory since it
yields images lacking crucial features of real mountains,
where some parts are smoother than others, due, for instance,
to erosion.

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
H(0,
1)with a
*regularity function*
.

mBm was introduced independently by two groups of authors: on the one hand, Peltier and Levy-Vehel defined the mBm from the moving average definition of the fractional Brownian motion, and set:

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

where denotes the complex white noise.

The Hölder exponents of the mBm are prescribed almost
surely: the pointwise Hölder exponent is
_{X}(
t) =
H(
t)
_{H}(
t)a.s., and the local Hölder
exponent is
a.s. Consequently, the regularity of the sample paths
of the mBm are determined by the function
Hor by its regularity. The multifractional Brownian
motion is our prime example of a stochastic process with
prescribed local regularity.

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
t, the Hölder exponent of the process
Xverifies almost surely
_{X}(
t) =
g(
X(
t)), where
gis a fixed deterministic function verifying certain
conditions. A process satisfying such an equation is
generically termed a
*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
(see section
). 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
B(
t,
H), where
(
t,
H)span
[0, 1]×[
a,
b]and
0<
a<
b<1. For each fixed
H,
B(
t,
H)is a fractional Brownian motion
with exponent
H. Denote:

the affine rescaling between
^{'}and
^{'}of an arbitrary continuous random field over a compact
set
K. One considers the following (stochastic) operator,
defined almost surely:

where
^{'}<
^{'}
,
and
are two real numbers, and
^{'},
^{'}are random variables adequately chosen.
One may show that this operator is contractive with respect
to the sup-norm. Its unique fixed point is the SRMP.
Additional arguments allow to prove that, indeed, the Hölder
exponent at each point is almost surely
g(
t).

An example of a two dimensional SRMP
with function
g(
x) = 1-
x^{2}is displayed on figure
.

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
(0,
2], the
*stability index*(
=
2corresponds to the Gaussian case, that we do not
consider here). This index measures in some precise sense the
intensity of jumps. Paths of stable processes with
close to 2 tend to display “small jumps”, while, when
is near 0, their aspect is governed by large ones.

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
is now a function of time. This allows to model
phenomena which, at times, are “almost continuous”, and at
others display large discontinuities. Such a behaviour is for
instance obvious on almost any sufficiently long financial
record.

More formally, a multistable process is a process which
is, at each time
u, tangent to a stable process
. Recall that a process
Yis said to be tangent at
uto the process
Y_{u}^{'}if:

where the limit is understood either in
finite dimensional distributions or in the stronger sense of
distributions. Note
Y_{u}^{'}may and in general will vary with
u.

One approach to defining multistable processes is similar
to the one developed for constructing mBm
: we consider fields of
stochastic processes
X(
t,
u), where
tis time and
uis an independent parameter that controls the
variation of
. We then consider a “diagonal” process
Y(
t) =
X(
t,
t), which will be, under certain
conditions, “tangent” at each point
tto a process
.

A particular class of multistable processes, termed “linear multistable multifractional motions” (lmmm) takes the following form , . Let be a -finite measure space, and be a Poisson process on with mean measure ( denotes the Lebesgue measure). An lmmm is defined as:

where
,
is a
C^{1}function and
and
are
C^{2}functions.

In fact, lmmm are somewhat more general than said above:
indeed, the couple
(
h,
)allows to prescribe at each point,
under certain conditions, both the pointwise Hölder exponent
and the local intensity of jumps. In this sense, they
generalize both the mBm and the linear multifractional stable
motion
. From a broader perspective,
such multistable multifractional processes are expected to
provide relevant models for TCP traces, financial logs, EEG
and other phenomena displaying time-varying regularity both
in terms of Hölder exponents and discontinuity structure.

**Multiparameter processes**

In order to use stochastic processes to represent the
variability of multidimensional phenomena, it is necessary to
define extensions for indices in
(
N2)
(see
for an introduction to the theory
of multiparameter processes). Two different kinds of
extensions of multifractional Brownian motion have already
been considered: an isotropic extension using the Euclidean
norm of
and a tensor product of one-dimensional processes on
each axis. We refer to
for a comprehensive survey.

These works have highlighted the difficulty of giving satisfactory definitions for increment stationarity, Hölder continuity and covariance structure which are not closely dependent on the structure of . For example, the Euclidean structure can be unadapted to represent natural phenomena.

A promising improvement in the definition of
multiparameter extensions is the concept of
*set-indexed processes*. A set-indexed process is a
process whose indices are no longer “times” or “locations”
but may be some compact connected subsets of a metric measure
space. In the simplest case, this framework is a
generalization of the classical multiparameter processes
: usual multiparameter processes
are set-indexed processes where the indexing subsets are
simply the rectangles
[0,
t], with
.

Set-indexed processes allow for greater flexibility, and should in particular be useful for the modeling of censored data. This situation occurs frequently in biology and medicine, since, for instance, data may not be constantly monitored. Censored data also appear in natural terrain modeling when data are acquired from sensors in presence of hidden areas. In these contexts, set-indexed models should constitute a relevant frame.

A set-indexed extension of fBm is the
first step toward the modeling of irregular phenomena within
this more general frame. In
, the so-called
*set-indexed fractional Brownian motion (sifBm)*was
defined as the mean-zero Gaussian process
such that

where is a collection of connected compact subsets of a measure metric space and .

This process appears to be the only set-indexed process
whose projection on increasing paths is a one-parameter
fractional Brownian motion
. The construction also provides
a way to define fBm's extensions on non-euclidean spaces,
*e.g.*indices can belong to the unit hyper-sphere of
. The study of fractal properties needs specific
definitions for increment stationarity and self-similarity of
set-indexed processes
. We have proved that the sifBm
is the only Gaussian set-indexed process satisfying these two
(extended) properties.

In the specific case of the indexing collection
, the sifBm can be seen as a multiparameter extension
of fBm which is called
*multiparameter fractional Brownian motion (MpfBm)*.
This process differs from the Lévy fractional Brownian motion
and the fractional Brownian sheet, which are also
multiparameter extension of fBm (but do not derive from
set-indexed processes). The local behaviour of the sample
paths of the MpfBm has been studied in
. The self-similarity index
His proved to be the almost sure value of the local
Hölder exponent at any point, and the Hausdorff dimension of
the graph is determined in function of
H.

The increment stationarity property for set-indexed processes, previously defined in the study of the sifBm, allows to consider set-indexed processes whose increments are independent and stationary. This generalizes the definition of Bass-Pyke and Adler-Feigin for Lévy processes indexed by subsets of , to a more general indexing collection. We have obtained a Lévy-Khintchine representation for these set-indexed Lévy processes and we also characterized this class of Markov processes.

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 (ANR and Pôle SYSTEM@TIC), and also through our involvement in the MASCOT-NUM research group (GDR of CNRS). In addition, we are considering probabilistic models as phenomenological models to cope with uncertainties in the DIGITEO ANIFRAC project. 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.

The design of a complex (mechanical) system such as aircraft, automobile or nuclear plant involves numerical simulation of several interacting physical phenomena: CFD and structural dynamics, thermal evolution of a fluid circulation, ... For instance, they can represent the resolution of coupled partial differential equations using finite element method. In the framework of uncertainty treatment, the studied “phenomenological model" is a chaining of different models representing the various involved physical phenomena. As an example, the pressure field on an aircraft wing is the result of both aerodynamic and structural mechanical phenomena. Let us consider the particular case of two models of partial differential equations coupled by limit conditions. The direct propagation of uncertainties is impossible since it requires an exploration and then, many calls to costly models. As a solution, engineers use to build reduced-order models: the complex high-fidelity model is substituted with a CPU less costly model. The uncertainty propagation is then realized through the simplified model, taking into account the approximation error (see ).

Interactions between the various models are usually explicited at the finest level (cf. Fig. ). How may this coupling be formulated when the fine structures of exchange have disappeared during model reduction? How can be expressed the interactions between models at different levels (in a multi-level modeling)? The ultimate question would be: how to choose the right level of modeling with respect to performance requirements?

In the multi-physical numerical simulation, two kinds of uncertainties then coexist: the uncertainty due to substitution of high-fidelity models with approximated reduced-order models, and the uncertainty due to the new coupling structure between reduced-order models.

According to the previous discussion,
the uncertainty treatment in a multi-physical and multi-level
modeling implies a large range of issues, for instance
numerical resolutions of PDE (which do not enter into the
research topics of
*Regularity*). Our goal is to contribute to the
theoretical arsenal that allows to fly among the different
levels of modelling (and then, among the existing numerical
simulations). We will focus on the following three axes:

In the case of a phenomenon represented by two coupled partial differential equations whose resolution is represented by reduced-order models, how to define a probabilistic model of the coupling errors? In connection with our theoretical development, we plan to characterize the regularity of this error in order to quantify its distribution. This research axis is supported by an ANR grant (OPUS project).

The multi-level modeling assumes the
ability to choose the right level of details for the
models in adequacy to the goals of the study. In order to
do that, a rigorous mathematical definition of the notion
of
*model fineness/granularity*would be very helpful.
Again, a precise analysis of the fine regularity of
stochastic models is expected to give elements toward a
precise definition of granularity. This research axis is
supported by a a Pôle SYSTEM@TIC grant (EHPOC project),
and also by a collaboration with EADS.

Some fine characteristics of the phenomenological model may be used to define the probabilistic behaviour of its variability. The action of modelling a phenomena can be seen as an interpolation issue between given observations. This interpolation can be driven by physical evolution equations or fine analytical description of the physical quantities. We are convinced that Hölder regularity is an essential parameter in that context, since it captures how variations at a given point induce variations at its neighbours. Stochastic processes with prescribed regularity (see section ) have already been used to represent various fluctuating phenomena: Internet traffic, financial data, ocean floor. We believe that these models should be relevant to describe solutions of PDE perturbed by uncertain (random) coefficients or limit conditions. This research axis is supported by a Pôle SYSTEM@TIC grant (CSDL project).

The problem, posed by Dassault Aviation, is that of
digital terrains assessment. Typically, several sets of
digital data are available for a single region. They
originate from different modalities (
*e.g.*radar images, geographical data, ...), have
different resolutions, and may be locally incomplete. The
challenge is to merge these data so as to obtain both a more
reliable description
*and*a “note” for each point,
*i.e.*a number assessing the confidence one has in this
particular value.

Our strategy is to model terrains with well-chosen stochastic processes, to estimate the parameters of the models, and then to use standard tools from statistics to qualify each point.

A first idea is to use mBm (which was precisely invented with this application in mind). More recently, we have used the SRMP as an alternative and sometimes more adapted model. Results using this approach are illustrated on figures and . They mainly show two facts:

On small enough zones, natural terrains do indeed exhibit a measurable relation between altitude and regularity, so that a modeling with a self-regulating process makes sense.

Estimation of the
gfunction of the SRMP indicates that young
mountains (such as Himalaya and the Rocky Mountains)
behave differently from older ones (such as Tibesti and
Massif Central): for young mountains, points at higher
altitudes are more irregular. The reverse seems to be
true for old mountains, possibly due to erosion
phenomena.

Our current work focuses on the search for better estimation methods of the parameters of the mBm modeling the terrains, a more thorough exploration of the relevance of SRP for terrain modeling, along with robust estimation methods, and finally on the development of an interpolation method based on local regularity, allowing to assess the quality of the available data.

**ECG analysis and modeling**

ECG and signals derived from them are an important source
of information in the detection of various pathologies,
including
*e.g.*congestive heart failure, arrhythmia and sleep
apnea. The fact that the irregularity of ECG bears some
information on the condition of the heart is well documented
(see
*e.g.*the web resource
http://

First, we use refined regularity characterizations, such as the regularization dimension, 2-microlocal analysis and advanced multifractal spectra for a more precise analysis of ECG data. This requires in particular to test current estimation procedures and to develop new ones.

Second, we build stochastic processes that mimic in a faithful way some features of the dynamics of ECG. For instance, the local regularity of RR intervals, estimated in a parametric way based on a modeling by an mBm, displays correlations with the amplitude of the signal, a feature that seems to have remained unobserved so far . In other words, RR intervals behave as SRP. We believe that modeling in a simplified way some aspects of the interplay between the sympathetic and parasympathetic systems might lead to an SRP, and to explain both this self-regulating property and the reasons behind the observed multifractality of records. This will open the way to understanding how these properties evolve under abnormal behaviour.

**Pharmacodynamics and patient drug compliance**

Poor adherence to treatment is a worldwide problem that
threatens efficacy of therapy, particularly in the case of
chronic diseases. Compliance to pharmacotherapy can range
from
5%to
90%. This fact renders clinical
tested therapies less effective in ambulatory settings.
Increasing the effectiveness of adherence interventions has
been placed by the World Health Organization at the top list
of the most urgent needs for the health system. A large
number of studies have appeared on this new topic in recent
years
,
. In collaboration with the
pharmacy faculty of Montréal university, we consider the
problem of compliance within the context of multiple dosing.
Analysis of multiple dosing drug concentrations, with common
deterministic models, is usually based on patient full
compliance assumption,
*i.e.*, drugs are administered at a fixed dosage.
However, the drug concentration-time curve is often
influenced by the random drug input generated by patient poor
adherence behaviour, inducing erratic therapeutic outcomes.
Following work already started in Montréal
,
, we consider stochastic
processes induced by taking into account the random drug
intake induced by various compliance patterns. Such studies
have been made possible by technological progress, such as
the “medication event monitoring system”, which allows to
obtain data describing the behaviour of patients.

We use different approaches to study this problem: statistical methods where enough data are available, model-based ones in presence of qualitative description of the patient behaviour. In this latter case, piecewise deterministic Markov processes (PDP) seem a promising path. PDP are non-diffusion processes whose evolution follows a deterministic trajectory governed by a flow between random time instants, where it undergoes a jump according to some probability measure . There is a well-developed theory for PDP, which studies stochastic properties such as extended generator, Dynkin formula, long time behaviour. It is easy to cast a simplified model of non-compliance in terms of PDP. This has allowed us already to obtain certain properties of interest of the random concentration of drug , . In the simplest case of a Poisson distribution, we have obtained rather precise results that also point to a surprising connection with infinite Bernouilli convolutions , . Statistical aspects remain to be investigated in the general case.

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 fifty
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 beginning to become
the case. To date, its use has been acknowledged in more than
120 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://

Our purpose is to build a stochastic calculus with respect
to mBm. We have first defined a stochastic integral with
respect to mBm in the frame of White Noise Theory developped
first by Hida. More precisely, we start from the normalized
mBm with functional parameter
hon
:

where
denotes a complex-valued Gaussian measure and where
for all
xin
(0, 1).

One approach to integration with respect to mBm is to use stochastic spaces in which one may actually differentiate stochastic processes such as Brownian motion. Considering the probability space where is given by Böchner Minlos theorem, White Noise Theory build two spaces, noted and which will play an analogous role to the space (the Schwartz space of rapidly decreasing functions which are infinitely differentiable) and (the space of tempered distributions).

We have shown that mBm
B^{(
h)}has the following Wiener-Itô chaos decomposition in
(
L
^{2}), the space of random variables defined
on the probability space
which admit a second order moment:

where
denotes the family of Hermite functions, defined for
every integer
kin
, by
e_{k}(
x): =
^{-1/4}(2
^{k}k!)
^{-1/2}e^{-
x
2/2}h_{k}(
x)and where
is the family of Hermite polynomial, defined for every
integer
kin
, by
. Note moreover that
M_{H}is an operator from
to
for every real
Hin
(0, 1)and
<.,
e
_{k}>is a centred random gaussian with second
moment order equal to 1 for all
kin
. We may then prove that the derivative of
B^{(
h)}in the sense of
exists and is equal to:

This leads to defining the integral with respect to mBm of any process as being the element of given by:

where denotes the Wick product on . It is then possible to obtain Itô formulas for functions with sub exponential growth and to solve stochastic differential equation driven by a mBm such as

*In collaboration with Prof. Kenneth Falconer (St Andrews
University, Scotland).*

We have pursued our studies of multistable processes,
introduced in
,
,
. We have obtained bounds on the
Hölder exponents of such processes, with an exact value in
the case of the Lévy multistalbe motion
. In this last situation the
exponent, is, at each
t, almost surely, equal to the value of the
localisability exponent, as expected. Obtaining uniform
results,
*e.g.*almost sure results for a path, leads to
performing a multifractal analysis, a task that we are
currently undertaking.

Another line of study is that of the estimation of the
functional parameters of multistable processes. We have
designed such estimators for the localisability and
regularity functions in the case of the Lévy multistable
motion and the the linear multifractional multistable motion
. Convergence in
L^{p},
p>0and almost sure convergence
have been proven.

*In collaboration with Prof. Ely Merzbach (Bar Ilan
University, Israel).*

For any integer
n, for all
and for all increasing sequences
and
in
,

This so-called
-increments
m-stationarity property is considered as the good
generalization of the increment stationarity property for
one-parameter processes since the projection of a stationary
process on any flow is one-parameter process with stationary
increments.

almost surely.

the increments of
Xare independent: for all pairwise disjoint
in
, the random variables
are independent.

Xhas
m-stationary
-increments, i.e. for all integer
n, all
and for all increasing sequences
(
U
_{i})
_{i}and
(
A
_{i})
_{i}in
, we have

Xis continuous in probability.

Contrarily to previous works of Adler and Feigin (1984) on one hand, and Bass and Pyke (1984) one the other hand, the increment stationarity property allows to characterize the distribution of a set-indexed Lévy process in terms of infinitely divisible probability measure (as in the real-parameter classical case).

If
is a set-indexed Lévy process and
such that
m(
U_{0})>0, then for all
the distribution of
X_{U}is equal to
(
P
_{XU0})
^{m(
U)/
m(
U0)}. Moreover the law of the Lévy
process
Xis completely determined by the law of
X_{U0}.

Conversely, for any infinitely divisible probability
measure
on
, there exists a set-indexed Lévy process
Xsuch that

This canonical representation of the set-indexed Lévy processes opens the door to a deep study : Characterization by an homogeneous Markov transition system, and a Lévy-Ito type representation (see for details).

*In collaboration with Benjamin Arras and Geoffroy
Barruel (students at Ecole Centrale Paris).*

The two classical ways to described the regularity of
stochastic processes, the local/pointwise Hölder exponents
and the fractal dimensions are connected in some specific
cases. For instance, if
is a real-valued fractional Brownian motion (fBm) with
self-similarity index
H(0,
1), the pointwise Hölder exponent at any point
satisfy
_{BH}(
t) =
Halmost surely. Besides, the
Hausdorff dimension of the graph of
B^{H}is given by
.

The connection between these two quantities can be interpreted as a consequence of an old paper from Adler (1977), who studied the local Hausdorff dimension of stationnary Gaussian fields.

In the present work, we studied a more general result, suitable for some larger class of processes.

which gives the almost sure value of the
local Hölder exponent of
Xat
t_{0}, we introduced a new exponent,
*deterministic local sub-exponent*

We proved the following result:

**Theorem (Pointwise almost sure result)**

If
, then the Hausdorff dimensions of the graph and the
range of
Xsatisfy almost surely,

We also proved the same kind of result for the Hausdorff
dimension of the range of
Xaround
t_{0}. Several extensions of this result have been stated:
an uniform almost sure result and a almost sure global
result.

As an application, we proved that if
is a multiparameter fractional Brownian motion with
index
H(0,
1/2], then with probability one, the Hausdorff
dimensions of the graph and the range of the sample paths of
are

We have defined a set-indexed extension of the well-known stationary Ornstein-Uhlenbeck process. This process (ssiOU) is defined as the zero-mean Gaussian process such that

where is a class of subsets of a measure metric space , satisfying some topological assumptions, and , are positive parameters.

We have defined a stationary property satisfied by this
process and similar to the definition of stationary
increments verified by the set-indexed fractional Brownian
motion. For any integer
n, for every
and for all increasing sequences
and
, we have

This -stationarity property extends the classic stationarity property for one-parameter processes. Furthermore, projections of -stationary processes along elementary flows are one-dimensional stationary processes.

We have also shown that the stationary set-indexed Ornstein-Uhlenbeck process verifies the following set-indexed Markov property, called -Markov property:

where
fis a bounded measurable function and
is the minimal filtration of
X.

Conversely, we get a characterization theorem similar to the one-parameter case, i.e., the stationary set-indexed Ornstein-Uhlenbeck process is the only zero-mean Gaussian process which has the following properties:

L^{2}inner- and outer-continuity;

-stationarity;

-Markov property.

Finally, we have shown that our definition extends the existing stationary multiparameter Ornstein-Uhlenbeck process defined in the literature. More precisely, this multiparameter process corresponds to the stationary set-indexed Ornstein-Uhlenbeck with the collection and the following measure:

where is the Lebesgue measure on and are the natural axes in .

Both the characterization result and the multiparameter case are two important justifications of the ssiOU's definition, which can lead to natural extensions of the Ornstein-Uhlenbeck process on non-Euclidian spaces.

*In collaboration with Prof. Florian de Vuyst (ENS
Cachan) and Antoine Merval (Ecole Centrale Paris).*

Computing solutions of a PDE problem parameterized by (N-dimensional) parameter requires the use of some Finite Elements (FE) code, which involves high computational times. The high cost in term of CPU time raises difficulties:

to perform robust analysis with respect to parameter ,

or, to optimize with respect to some criterion .

An usual way to deal with the CPU cost is to build
reduced-order representations of
. In non-stationary cases, a snapshot of
at some fixed time
tmay not be as interesting as the study of some
integrated values such as mean during some time interval or
proportion of time spent over or below some threshold. The
formalism of probability is particularly adapted to such a
description: considering time variable as the alea leads to
view integrated quantities in time as moments of a random
variable or quantiles.

More precisely, in order to describe the evolution of
along the time interval
[0, 1], our idea is to consider the
uniform probability measure on
[0, 1]and to identify
for
=
t= [0, 1]. Then the integrated
quantities can be interpreted as the expectation
, the variance
, or the probabilities
, and the exploration of the parameter space or the
interpolation between some given values of
is realized by the analysis of the multiparameter
stochastic process
. In this work, we restrict to Gaussian processes
whose covariance structure, i.e. the functions
and
, is parametrized by the local regularity of the
quantity
with respect to
. We think that the multifractional processes, which
have been studied by Regularity members for several years,
can provide a model of probabilistic interpolation between
fixed values of
.

This approach is illustrated on a database made of time-series representing temperature evolution in an aircraft cabin, computed by a FE model ruled by Navier-Stokes equations. These works have been submitted to publication .

In the frame of ANIFRAC, we have tried to use SRP to
detect arrhythmia from RR intervals, and to assess the
efficiency of certain drugs. The basic idea is as follows: if
ECG are well modeled with self-regulating processes, it seems
plausible that arrhythmias will modify the dynamics of the
relation between the amplitude of the signal and its local
regularity. Such changes should be noticeable on estimations
of the
gfunction of the self-regulating process, providing
detection of such events. Our results so far indicate that it
is indeed possible to detect with high accuracy patients
suffering from arrhythmia by analyzing their
gfunction. This allows in turn to quantify the effect
of certain drugs
.

*In collaboration with P.E. Lévy Véhel (University of
Nice-Sophia-Antipolis and Banque Postale)*

The purpose of a multiple-dosing regimen is to achieve and maintain a consistent pharmacological response for a period longer than the duration of a single dose administration. Practically, a loading dose is given to quickly achieve a quasi-steady-state concentration level, followed by maintenance doses to keep the concentration within this level. We consider the classical multiple intravenous (multi-IV) and multiple oral (multi-oral) models with the simple one-compartment pharmacokinetic model and the first-order kinetics.

**Concentration response in the multi-IV case**

The concentration in the multi-IV case can be written in
the following way. Assume that a patient takes doses
D_{i}at times
T_{i}. These doses translate into immediate (
*i.e.*at each time
T_{i}) increases of the concentration by the value
, where
V_{d}is the volume of distribution. After that, the effect
of the dose taken at
T_{i}on the overall concentration decreases exponentially
fast, with exponential speed
k_{e}. Formally, the concentration is given by

**Concentration response in the multi-oral case**

The multiple oral doses model considers two important
processes, the first is the oral absorption process defined
by the amount of drug at the absorption site remaining to be
absorbed, which is characterized by the absorption
coefficient rate
k_{a}. The other is the elimination process defined by the
irreversible loss of drug from the site of measurement, which
is eliminated with a rate constant
k_{e}. Thus, we have the following expression to the drug
concentration process,

where
Fis the absolute bioavailability and
V_{d}is the apparent volume of distribution.

**Variability and singularity arising from poor compliance
in a PD/Pk model**

In the seminal works , , the authors attacked this using a probabilistic frame. Our work is similar in spirit. In our first article , in a series of three, we consider models of increasing generality and complexity. We investigate the probability distribution of drug concentration in the context of multiple-IV dosing and poor compliance. In a second article , we consider the more realistic multi-oral model and poor compliance. We suppose that the moments of drug intake to follow a Poisson process. This assumption allows to obtain precise results describing various aspects of the distribution of the concentration that are important for assessing the efficacy of the regimen.

In the multiple oral dosing case, the characteristic function is given by

We focus on aspects of practical
relevance: the
*variability*of the concentration, the
*regularity*of its probability distribution and of its
limit probability distribution. It is intuitively obvious
that poor compliance will increase the variability of the
concentration around its mean as compared to the full
compliance case. Our results quantify this in a precise way,
showing the exact role played by each parameter of the
process.

An even more radical situation occurs if, instead of considering a continuous time model, one investigates another approach to the modeling of drug concentration by analyzing a time-discretized version: in this setting, the problem at hand reveals unexpected links with possibly multifractal measures. Again, depending on some parameters, the discretized concentration may exhibit an extremely irregular behavior. This is obviously an undesirable feature which may have strongly negative consequences.

**Modeling patient poor compliance in the multi-IV
administration case with Piecewise Deterministic Markov
Models**

We use a particular piecewise deterministic Markov process (PDMP) to model the drug concentration in the multi-IV case . The model allows to take into account the irregular drug intake times. We study the stochastic properties of the PDMP through its infinitesimal generator given by

*In collaboration with Dassault Aviation, EDF, EADS.*

A general methodology has been defined to manage uncertainties in the numerical simulation context. An intensive collaboration with R&D entities of industrial compagnies led to a common view of the problem.

At the early stage of aircraft design, the models involved are very simplified and the geometric and environmental variables are not completely determined. Then, the prescribed performances of the designed aircraft are uncertain and considered as random variables. In , , , , the general issue of robust aircraft design has been stated in terms of probability framework and multi-disciplinary optimization of uncertain variables.

The preliminary design of complex systems is common to several areas, such as aeronautics, automobile and energy industries. It can be described as an exploration process of a so-called design space, generated by the global parameters. An interactive exploration, with a decisional visualization goal, needs reduced-order models of the involved physical phenomena. We are convinced that the local regularity of phenomena is a relevant quantity to drive these approximated models. Roughly speaking, in order to be representative, a model needs more informations where the fluctuations are the more important (and consequently, where irregularity is the more important).

In collaboration with Dassault Aviation, EDF and EADS, we
study how the local regularity can provide a good
quantification of the concept of
*granularity*of a model, in order to select the good
level of fidelity adapted to the requiered precision.

A particular aspect of our works in that field is the study of the evolution of the local regularity inside partial differential equations (PDE), such as models coming from fluid dynamics. The fluctuating phenomena is represented by stochastic processes with prescribed regularity, and the knowledge of the fine behaviour of the solution of the PDE will provide important informations in the view of numerical simulations.

EHPOC project of the Pôle de Compétitivité SYSTEM@TIC PARIS-REGION (ended in 08/2010). The industrial partners involved were CEA, Dassault Aviation, EADS, EDF. The goal of the project was the development of a generic methodology to manage uncertainties and its demonstration through industrial cases.

CSDL (Complex Systems Design Lab) project of the Pôle de Compétitivité SYSTEM@TIC PARIS-REGION (11/2009-10/2012). Among the involved industrial partners, we can mention Dassault Aviation, EADS, EDF, MBDA and Renault. The goal of the project is the development of a scientific plateform of decisional visualization for preliminary design of complex systems.

EHPOC project of the Pôle de Compétitivité SYSTEM@TIC PARIS-REGION. The academic partners involved in the uncertainty workpackage were ECP (Prof. Florian de Vuyst) and INRIA Select (Gilles Celeux).

CSDL project of the Pôle de Compétitivité SYSTEM@TIC PARIS-REGION. The academic partners involved include ECP, Ecole des Mines de Paris, ENS Cachan, INRIA, Supelec.

DIGITEO ANIFRAC project on uncertainties management in pharmacodynamics and ECG anlysis. The involved academic partners include ECP, INRIA, Supelec and Nantes University.

Jacques Lévy Véhel was an invited speaker at the “Workshop on Fundamental Research Problems of the Future Internet” held in Budapest in September.

The Regularity team collaborates with Bar Ilan university on theoretical developments around set-indexed fractional Brownian motion and set-indexed Lévy processes (invitations of Erick Herbin in Israël during four months in 2006, 2007, 2008 and 2009 and invitation of Prof. Ely Merzbach at Ecole Centrale Paris in 2008, 2009 and 2010).

The Regularity team collaborates with Michigan State University (Prof. Yimin Xiao) on the study of fine regularity of multiparameter fractional Brownian motion (invitation of Erick Herbin at East Lansing in 2010).

Erick Herbin was invited to School of Mathematics Seminar (Georgia Tech, Atlanta, USA) in May, 2010.

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

The Regularity team collaborates with Acadia University (Canada, Prof. Franklin Mendivil) on the study of multifractal strings.

Erick Herbin is involved in the organization of the continuing education program "Engager et élaborer une démarche incertitudes", under the labels IMdR (Institut de Maitrise des Risques), SMAI (Société de Mathématiques Appliquées et Industrielles), SFdS (Société Française de Statistiques) and TERATEC.

Erick Herbin is member of the IMdR Work Group "Uncertainty and industry".

Erick Herbin is member of the CNRS Research Group GDR Mascot Num, devoted to stochastic analysis methods for codes and numerical treatment.

Erick Herbin is reviewer for Mathematical Reviews (AMS).

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

Jacques Lévy Véhel was a reviewer for the Ph.D. thesis of Hédi Kortas and a reviewer for the Habilitation à Diriger des Recherches of C. Gentil.

Erick Herbin is in charge of the Probability Course at Ecole Centrale Paris (20h).

Erick Herbin is in charge of the Random Modeling Course at Ecole Centrale Paris (30h).

Erick Herbin and Jacques Lévy Véhel are in charge of the Brownian Motion and Stochastic Calculus Course at Ecole Centrale Paris (30h).

Erick Herbin gives travaux dirigés on Real and Complex Analysis at Ecole Centrale Paris (10h).

Erick Herbin is in charge of the Numerical Simulation Program in the Applied Mathematics option of Ecole Centrale Paris.

Erick Herbin is supervisor of several student's research projects in the field of Mathematics at Ecole Centrale Paris.

Paul Balança gives travaux dirigés on Probability, Real and Complex Analysis at Ecole Centrale Paris (20h).

Paul Balança gives travaux dirigés on Random Modeling at Ecole Centrale Paris (17h).

Jaochim Lebovits gives travaux dirigés on analysis and probability at Ecole Centrale Paris (12h).

Jaochim Lebovits gives travaux dirigés on financial mathematics at Ecole Centrale Paris (6h).

Jaochim Lebovits supervises students research projects on financial mathematics at Ecole Centrale Paris.

Jaochim Lebovits gives travaux dirigés on stochastic calculus at Ecole Centrale Paris (15h).

Lisandro Fermin gives travaux dirigés on stochastic modeling mathematics at Ecole Centrale Paris (9h).

Lisandro Fermin gives travaux dirigés on probability at Ecole Centrale Paris (9h).

Lisandro Fermin gives travaux dirigés on statistics at University Paris X- Nanterre (18h).