Blanc SIMI 1 - Sciences de l'information, de la matière et de l'ingénierie : Mathématiques et interactions

MAlliavin, STEin, Random Irregular Equations – MASTERIE

MASTERIE

Malliavin, Stein, Random Irregular Equations

General objectives

We wish to expand the Stein/Malliavin approach beyond the<br />framework of Gaussian (or Gamma) approximations - as developed in a recent series of papers by Nourdin, Peccati and coauthors.<br />One of our principal aims is to develop a unified approach to<br />central and non-central phenomena appearing in several branches of stochastic analysis. For instance, we are interested in central and non-central limit theorems emerging in the study of<br />power variations of fractional Gaussian processes. Random<br />matrix theory represents another exciting domain where to apply<br />our findings. Another exciting venue for our<br />research is the extension of the Stein/Malliavin approach to a<br />functional framework \`a la Donsker. This type of results would<br />fill an important gap in the existing literature: they are also<br />tightly related with the use of Malliavin-based techniques in<br />order to derive concentration inequalities, with specific<br />emphasis on the evaluation of small-ball probabilities<br />associated with a Gaussian process on the real line.<br /><br />We wish to put the basis of performing analysis for some<br /> stochastic evolution problems<br />with very irregular coefficients. Among those we mention three<br />peculiar cases.<br />1) One example is given by a parabolic PDE with noisy<br />time-dependent drift.<br />2) A class of PDE of non-linear monotonic type with<br />possibly discontinuous coefficients and its microscopic<br />probabilistic interpretation. Those equations<br />are motivated by some complex systems modeling.<br /><br />3) To put the basis of a stochastic calculus via<br />regularisation in Banach spaces, in particular<br />for processes with values in $C[[-\tau,0]$ where<br />$\tau > 0$ and where the integrator processes are not<br />semimartingales. This will have as a side-effect<br />a representation formula for random variables<br />living in a non probability space based on a<br />finite quadratic variation process.

The tasks 1 and 2 are relatively independent
with respect to tasks 2 and 3.
During the first semester, there was
a meeting with some talks of the participants
and several other guests.
This has impulsed a precise program.
During the first two years, the two groups of tasks
will run independently, with frequent small meetings
in order to exchange ideas.
Some conferences are coorganized.
A fruitful interaction will take place with
the Semester «Stochastic analysis and related fields« of the
Bernoulli Center of EPFL Lausanne. F. Russo is
coorganizer. At that occasion, short courses
for young scientists were organized.
During the second year a new meeting of hte
participants will take place and the different
tasks should merge.

The research has advanced with regularity according to the program. A striking result is the prize obtained by I. Nourdin,
by the Fondation des Sciences Mathématiques de Paris.

Applications in statistics, partial differential equations and numerics

Articles published in the top journals of the field.

The project consists in exploring some new applications of
Malliavin calculus (or more general infinite-dimensional
techniques) to attack two classes of problems. One one side, we
look for new developments in Stein's method, on the other hand,
we wish to develop new tools of calculus which, together with
regularization techniques, allow to solve some specific Random
Irregular Equations: among those, we are interested in
time-dependent stochastic differential equations with
generalized drift and perturbations of non-linear PDEs with
monotone singular coefficients. Concerning Stein's method, we
will direct our efforts towards the following main goals: (i)
the extension of the existing techniques (based on the
interaction of Stein's method and Malliavin calculus) in order
to encompass non-Gaussian and non-Gamma approximations; (ii)
the derivation of infinite-dimensional concentration
inequalities, with special focus on small-ball probabilities;
(iii) a full Malliavin-based development of a functional
version of Stein's method in the framework of approximation
results concerning random elements taking values in a
functional space. One should notice that the use of
Malliavin/Stein techniques in order to tackle Points (i)-(iii)
is completely original, and would provide a great number of new
insights into probabilistic approximations associated with an
infinite-dimensional framework.

Project coordination

Francesco Russo (UNIVERSITE PARIS 13) – Francesco.Russo@ensta-paristech.fr

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

UP13 UNIVERSITE PARIS 13
UPS UNIVERSITE TOULOUSE III [PAUL SABATIER]
USTL UNIVERSITE DE LILLE I [SCIENCES ET TECHNOLOGIES]

Help of the ANR 155,599 euros
Beginning and duration of the scientific project: - 36 Months

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