What is it?
- The aim of this three-day workshop is to share our enthusiasm for the rough paths theory and the stochastic analysis related to fractional fields, by bringing together many of the mathematicians interested in these topics, in order to report on the newest developments and to initiate further joint project.
- The organizers are Céline Lacaux, Ivan Nourdin and Samy Tindel.
- The workshop is generously supported by the Agence Nationale de la Recherche through the two projects "Exploration of Rough Paths" (ECRU) and "Malliavin, Stein and Stochastic Equations with Irregular Coefficients" (MASTERIE)
- The acronym `STAN DAYS' is for STochastic ANalysis DAYS, but also because the Place Stan', which belongs in the list of UNESCO World Heritage Sites, is the most famous place in Nancy.
Practical informations
- The list of speakers is composed of:
Hermine Biermé (Paris 5 and Tours),
Serge Cohen (Toulouse),
Aurélien Deya (Nancy),
Massimiliano Gubinelli (Paris 9),
Martin Hairer (Warwick),
Christian Litterer (Imperial College),
Andreas Neuenkirch (Mannheim),
David Nualart (Kansas),
Anastasia Papavasiliou (Warwick),
Giovanni Peccati (Luxembourg),
Sebastian Riedel (Berlin),
Mark Podolskij (Heidelberg),
Murad Taqqu (Boston),
Ciprian Tudor (Lille),
Frederi Viens (Purdue)
and
Lorenzo Zambotti (Paris 6).
-The list of participants can be found here 
-By clicking here, you will find the pictures of the conference 
- The poster of the workshop can be downloaded here
Program (see below for the abstracts of the talks)
Talks were 40 minutes long + 5 minutes for questions.
May 9th
13:50 Introduction to the workshop
Session ``Stochastic Analysis'' (Chairman: Frederi Viens)
14:00 David Nualart (Kansas): ``Central limit theorem for additive functionals of the fractional Brownian motion'' [PDF file]
14:45 Giovanni Peccati (Luxembourg): ``A Poisson/Gaussian alternative on configuration spaces'' [PDF file]
15:30 Coffee break
16:00 Aurélien Deya (Nancy): ``q-Brownian motion and the Fourth Moment Theorem'' [PDF file]
16:45 Ciprian Tudor (Lille): ``Stein's method for invariant measures of diffusions via Malliavin calculus'' [PDF file]
17:30 End of the first day
May 10th
Session ``Rough paths theory'' (Chairman: Murad Taqqu)
9:00 Martin Hairer (Warwick): ``Solving the KPZ equation'' [PDF file]
9:45 Christian Litterer (Imperial College): ``Integrability and tail estimates for SDEs driven by Gaussian noises and applications''
10:30 Coffee break
11:00 Massimiliano Gubinelli (Paris Dauphine): ``Some applications of controlled paths'' [PDF file]
11:45 Sebastian Riedel (TU Berlin): ``Rates of Convergence of Wong-Zakai Approximations for SDEs driven by Gaussian signals'' [PDF file]
12:30 Lunch
Session ``Fractional fields and SPDEs'' (Chairman: David Nualart)
14:00 Frederi Viens (Purdue): ``Extensions of inequalities for Gaussian fields via the Malliavin calculus'' [Whitebooard presentation]
14:45 Murad Taqqu (Boston University): ``Long-range dependence and the rank of decompositions'' [PDF file]
15:30 Coffee break
16:00 Serge Cohen (Toulouse): ``Approximation of stationary solutions of Gaussian driven Stochastic Differential Equations'' [PDF file]
16:45 Lorenzo Zambotti (Paris 6): ``CSBPs with immigration and Fleming-Viot Processes with Mutation'' [PDF file]
17:30 End of the second day, followed by a dinner in a local restaurant all together
May 11th
Session ``Statistical aspects'' (Chairman: Giovanni Peccati)
9:00 Anastasia Papavasiliou (Warwick): ``Statistical Inference for Rough Differential Equations'' [PDF file]
9:45 Hermine Biermé (Paris 5 and Tours): ``Breuer Major Theorem revisited'' [PDF file]
10:30
Coffee break
11:00 Andreas Neuenkirch (Mannheim): ``A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise'' [Whiteboard presentation]
11:45 Mark Podolskij (Heidelberg): ``Some asymptotic results on semi-stationary Lévy processes'' [Whiteboard presentation]
12:30 Lunch
14:00 End of the third day and of the workshop
Hermine Biermé (MAP5 Université Paris Descartes and LMPT Université de Tours) Breuer Major Theorem revisited
The Breuer Major Theorem states a Central Limit Theorem for normalized sums of a functional of a Gaussian stationary time-series, under a summability assumption of the covariance function. Based on recent results on the Stein's method for normal approximation using Malliavin calculus, a new proof allows to refine this theorem, by giving the rate of convergence in some cases. Some of these results also extend to triangular arrays. Partially based on joint works with Aline Bonami, Ivan Nourdin, Giovanni Peccati and José R. León. Serge Cohen (Université de Toulouse) Approximation of stationary solutions of Gaussian driven Stochastic Differential Equations
We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent of the future of the underlying innovation process of the Gaussian driving process. Aurélien Deya (Université de Lorraine) q-Brownian motion and the Fourth Moment Theorem We extend the Fourth Moment Theorem of Nualart and Peccati to
the so-called q-Brownian motion (q in (0,1)) introduced by Frisch and
Bourret in the 70's. The main part of the talk will actually be devoted to
the presentation of the q-Brownian motion, which interpolates between
the standard Brownian motion (q=1) and the free Brownian motion (q=0),
and is one of the nicest examples of non-commutative process. Accordingly,
some preliminaries on non-commutative probability theory will be provided. Massimiliano Gubinelli (Paris Dauphine) Some applications of controlled paths
Controlled paths have been introduced to provide an alternative formulation of the rough path theory of Lyons. I will illustrate some applications of the idea of controlled paths in contexts unrelated to the integration theory for stochastic processes. In particular in PDEs and in the phenomenon of regularization by noise for ordinary differential equations. Martin Hairer (The University of Warwick) Solving the KPZ equation
The KPZ equation was originally introduced in the eighties as a model of
surface growth, but it was soon realised that its solution is a "universal" object
describing the crossover between the Gaussian universality class and the KPZ
universality class. The mathematical proof of its universality however is still an
open problem, in particular because of the lack of a good approximation theory
for the equation. Indeed, the only known way so far to mathematically interpret solutions
to the KPZ equation is to reduce it to a linear stochastic PDE via a non-linear
transformation called the Cole-Hopf transform. Unfortunately, the resulting linear
equation does itself lack a good approximation theory and many microscopic models
do not behave well under the Cole-Hopf transform. Christian Litterer (Imperial College, London) Integrability and tail estimates for SDEs driven by Gaussian noises and applications
We derive explicit tail-estimates for the Jacobian of the solution
flow of stochastic differential equations driven by Gaussian rough paths. In
particular, we deduce that the Jacobian has finite moments of all order for
a wide class of Gaussian process including fractional Brownian motion with
Hurst parameter H>1/4. Finally, we consider an application in a Hormander
type result for the smoothness of the density in such equations. Andreas Neuenkirch (Universität Mannheim) A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise We study a least square-type estimator for an unknown parameter in the drift coefficient of a
stochastic differential equation with additive fractional noise of Hurst parameter H>1/2. The estimator is based on discrete time observations of the stochastic differential equation,
and using tools from ergodic theory and stochastic analysis we derive its strong consistency. David Nualart (The University of Kansas) Central limit theorem for additive functionals of the fractional Brownian motion In this talk we will present a central limit theorem for an additive functional of the d-dimensional fractional Brownian motion with Hurst index H in
(1/(d+1),1/d), extending a result by Papanicolaou, Stroock and Varadhan in the case of the standard Brownian motion. The limit is a Brownian motion with a random variance which involves the local time of the fractional Brownian motion at the origin. The proof is done using the method of moments. This is a joint work with Yaozhong Hu and Fangjun Xu. Anastasia Papavasiliou (The University of Warwick) Statistical Inference for Rough Differential Equations I will discuss how to estimating unknown parameters in the vector fields of an RDE given discretely observed realizations of the rough path. I first discuss a moment-matching type of estimator and then a maximum likelihood estimator. I will also discuss applications.
Giovanni Peccati (Luxembourg University) A Poisson/Gaussian alternative on configuration spaces We shall describe a class of analytic inequalities on
configuration spaces, allowing to assess normal and Poisson
approximations of functionals of random measures. Strong motivations
come from stochastic geometry - in particular from the theory of
geometric U-statistics and random graphs. Partially based on joint
works with Raphaël Lachièze-Rey. Mark Podolskij (Universität Heidelberg) Some asymptotic results on semi-stationary Lévy processes In this talk we present some new asymptotic results on high frequency functionals of semi-stationary Lévy processes.
Semi-stationary Lévy processes (SSL), which are a subclass of the so called ambit processes, constitute models for
turbulence in physics. The aim of the talk is to present a picture of limits which may appear if one considers
high frequency statistics of SSL processes, such as power variations. We show some laws of large numbers and the corresponding
stable central limit theorems. Sebastian Riedel (TU Berlin) Rates of Convergence of Wong-Zakai Approximations for SDEs driven by Gaussian signals We briefly review the structural conditions for a multi-dimensional Gaussian process to yield a naturally associated (random) rough path in the sense of Lyons. The resulting stochastic differential equations (better: random rough differential equations) have been subject to intense investigations, especially in the case of fractional Brownian motion (fBM). Our contribution, in a general Gaussian rough path context which includes fBM with H > 1/4, is to establish somewhat definite a.s. rates of convergence, both for Wong-Zakai and ``simplified'' Milstein-type approximations, sharpening previous results of Hu-Nualart, Deya-Neuenkirch-Tindel. Under the additional assumption of complementary Young regularity of the Cameron-Martin space, also satisfied for fBM with H>1/4, we can adapt ideas of Cass-Litterer-Lyons to obtain also rates in Lq, any q<∞. Murad Taqqu (Boston University) Long-range dependence and the rank of decompositions We review and compare different methodologies for studying the asymptotic behavior of partial sums of nonlinear functionals of the following type
h(X1)+...+h(XN) in the long-range dependence setting.
Here (X1,X2,...) is either a stationary
mean-zero Gaussian process or a
linear process. The methodologies, we consider, are based on different
decompositions of the function h. This includes the decomposition of
Surgailis (1982) and of Ho and Hsing (1997) in the case of
linear processes. The so-called ``rank'' of these
decompositions plays an essential role. We show
that all ranks coincide when the function h is a polynomial. Ciprian Tudor (Lille University) Stein's method for invariant measures of diffusions via Malliavin calculus Given a random variable F regular enough in the sense of the Malliavin calculus, we measure the distance between its law and any probability measure with a density function which is continuous, bounded, strictly positive on an interval in the real line and admits finite variance. The bounds are given in terms of the Malliavin derivative of F. Our approach is based on the theory of
Itô diffusions and the stochastic calculus of variations and extends certain results obtained by Nourdin and Peccati. Several examples are considered in order to illustrate our general results. Frederi Viens (Purdue University) Extensions of inequalities for Gaussian fields via the Malliavin calculus We will discuss recent developments in the application of the Malliavin calculus, initiated by Nourdin and Peccati, to provide quantitative estimates for random variables and fields on Wiener space. In particular, we will see how the notion of Gaussian distribution and of covariance can be generalized by using Malliavin calculus operators, which are then used to extend classical theorems to non-Gaussian fields, including the Dudley-Fernique entropy and the Fernique-Sudakov comparisons for expected suprema, concentration in the vein of Poincare inequalities, and Gordon-Slepian inequalities. We will mention applications to polymers in random environments, and to the Sherrington-Kirkpatrick model. This represents joint work in progress with Ivan Nourdin and Giovanni Peccati. Lorenzo Zambotti (LPMA, Université Pierre et Marie Curie, Paris) CSBPs with immigration and Fleming-Viot Processes with Mutation We study a class of self-similar jump type SDEs driven by
Hölder continuous drift and noise coefficients. Using the Lamperti
transformation for positive self-similar Markov processes we obtain a
necessary and sufficient condition for almost sure extinction in
finite time. We then show that in some cases pathwise uniqueness
holds in a restricted sense, namely among solutions spending a
Lebesgue-negligible amount of time at 0. We then study an
associated Fleming-Viot process with mutation. (Joint work
with J. Berestycki, L. Doering and L. Mytnik).Abstracts
In this talk, we present a new notion of solution to the KPZ equation that
bypasses the use of the Cole-Hopf transform. Our approach also allows to
factorise the solution map into a "universal" (i.e. independent of initial condition)
measurable map, composed with a solution map with good continuity properties.
This lays the foundations for a robust approximation theory to the KPZ equation,
which is needed to prove its universality. As a byproduct of the construction, we
obtain very detailed regularity estimates on the solutions, as well as a new
homogenisation result.
Joint work with Peter Friz (Berlin) and Weijun Xu (Oxford).
This is joint work with Céline Lévy-Leduc.