STOCHASTIC
SYSTEMS IN MATHEMATICS AND MATHEMATICAL PHYSICS

ANR project in the category Programme Blanc-2011

Teams and members Publications Conferences Position Abstract

ANR project in the category Programme Blanc-2011

Teams and members Publications Conferences Position Abstract

Teams
and members

The project involves three teams and forteen researchers:

ENS de Cachan, Antenne de Bretagne : R. Belaouar, A. de Bouard, A. Debussche (coordinator of Cachan's team), E. Gautier, J. Vovelle

Université de Cergy-Pontoise : M. Kleptsyna, S. Kuksin, Y. Le Jan, V. Nersesyan, A. Shirikyan (coordinator of the project), A. Popier

ENS de Lyon : F. Bouchet, R. Chetrite, K. Gawedzki (coordinator of Lyon's team)

Publications

More than 60 articles were published with a partial support of the project.

Conferences

The ANR project STOSYMAP participated in organisation of a number of conferences and workshops.

Position

From March of 2013 to August of 2014, Alberto Maiocchi held an 18-month postdoc position shared between the University of Cergy-Pontoise and Institut de Mathématiques de Jussieu.

Abstract of the project

We will consider mathematical issues related to solutions of the 3D (three-dimensional) Navier–Stokes equations (the classical turbulence setup) with high Reynolds numbers, and various statistical characteristics of these solutions. We will also consider simpler related models: the two-dimensional stochastic Navier–Stokes equations (2D turbulence, relevant for meteorology and some fields of physics), the stochastic nonlinear Schrödinger equation in dimensions 1, 2, and 3 (optical turbulence), the Gross–Pitaevskii equation with stochastic perturbations (turbulence in Bose–Einstein condensation), the stochastic Burgers equation (a popular toy model for the classical turbulence), the Korteweg–de Vries equation with small dissipation and random force (another physical model for turbulence in various media).

The project is formed of analytical researches of qualitative properties of solutions for the equations above and for other similar problems. They are supported by numerical studies of the corresponding models. More specifically, our team plans to consider the following questions:

In parallel to the above-mentioned questions, we will investigate the more challenging (and unpredictable) problem of qualitative behavior of solutions for the 3D Navier–Stokes system in bounded and unbounded domains. Only few mathematical results related to the phenomenon of turbulence are known in this context, and there is no good understanding of the problem on the physical level of rigor. Our program will include the investigation of following problems for the 3D Navier–Stokes system and other related and/or simplified equations:

The first aim (and main cost) of this four-year project is to extend the existing research effort, by hiring post-docs to work with the involved researchers. The project will also develop long-term relationships between the partners laboratories, each of them internationally recognized in their own field. Finally, the project will fund workshops and meetings to foster international collaborations and discussions.

The project involves three teams and forteen researchers:

ENS de Cachan, Antenne de Bretagne : R. Belaouar, A. de Bouard, A. Debussche (coordinator of Cachan's team), E. Gautier, J. Vovelle

Université de Cergy-Pontoise : M. Kleptsyna, S. Kuksin, Y. Le Jan, V. Nersesyan, A. Shirikyan (coordinator of the project), A. Popier

ENS de Lyon : F. Bouchet, R. Chetrite, K. Gawedzki (coordinator of Lyon's team)

Publications

More than 60 articles were published with a partial support of the project.

Publications in refereed journals

- Albeverio S., Debussche A., Xu L., Exponential mixing of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises, Applied Mathematics and Optimization 66 (2012), no. 2, 273–308.
- Alouges A., de Bouard A., Hocquet A., A semi-discrete scheme for the stochastic Landau-Lifshitz equation, Stochastic Partial Differential Equations: Analysis and Computations (2014), à paraitre.
- Ammari K., Duyckaerts T., Shirikyan A., Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation, hal-00759119 (30/11/2012).
- Aurell E., Gawedzki K., Mejia-Monasterio C., Mohayaee R., Muratore-Ginanneschi P., Refined Second Law of Thermodynamics for fast random processes, J. Stat. Phys. 147 (2012), 487–505.
- Barato A.C. , Chetrite R., Hinrichsen H. , Mukamel D. A Gallavotti-Cohen-Evans-Morriss like symmetry for a class of Markov jump processes, J. Stat. Phys 146 (2012), no. 2, 294–313.
- Barato A.C., Chetrite R., On the symmetry of current probability distri-butions in jump processes, J. Phys. A: Math. Theor. 45 (2012), 485002.
- Barré J., Chetrite R., Muratori M., Peruani P., Motility-induced phase separation of active particles in the presence of velocity alignment, J. Stat. Phys. (2014), à paraître.
- Bauer M., Chetrite R., Ebrahimi-Fard K., Patras F., Time ordering and a generalizes magnus expansion, Lett. Math. Phys. 103 (2012),no. 3, 331–350.
- Belaouar R., de Bouard A., Debussche A., Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion, hal-00948570 (2013).
- Bouchet F., Touchette H., Non-classical large deviations for a noisy system with non-isolated attractors, J. Stat. Mech. (2012), P05028.
- Bouchet F., Venaille A., Statistical mechanics of two-dimensional and geophysical flows, Physics Reports 515 (2012), no. 5, 227–295.
- Bouchet F., Nardini C., Tangarife T., Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier-Stokes equations, J. Stat. Phys. 153 (2013), no. 4, 572–625.
- Bouchet F., Laurie J., Zaboronski O., Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional Euler equations, J. Stat. Phys (2014), à paraître.
- Cai C., P. Chigansky P., Kleptsyna M., Mixed fractional Brownian motion: the filtering perspective, arXiv:1208.6253 (2012).
- Chetrite R., Mallick K., Quantum fluctuation relations for the Lindblad master equation, J. Stat. Phys. 148 (2012), 480–501.
- Chetrite R., Touchette U., Nonequilibrium microcanonical and canonical ensembles and their equivalence. Phys. Rev. Lett 111, (2013) 120601.
- Chetrite R., Touchette U., Nonequilibrium Markov processes conditioned on large deviations, arXiv :1405.5157 (2014).
- Corvellec M., Bouchet F., A complete theory of low-energy phase diagrams for two-dimensional turbulence steady states and equilibria, arXiv:1207.1966 (2012).
- de Bouard A., Fukuizumi R., Representation formula for stochastic Schrödinger evolution equations and applications, Nonlinearity 25 (2012), no. 11, 1993–3022.
- de Bouard A., Gazeau M., A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers, Annals of Applied Probability 22 (2012), 2460–2504.
- Debussche A., Vovelle J., Diffusion limit for a stochastic kinetic problem, Communications on Pure and Applied Analysis 11 (2012), no. 6, 2305–2326.
- Debussche A., Glatt-Holtz N., Temam R., Ziane M., Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise, Nonlinearity 25 (2012), 2093.
- Debussche A., Romito M., Existence of densities for the 3D Navier-Stokes equations driven by Gaussian noise, Probability Theory and Related Fields 158 (2014), no. 3-4, 575–596.
- Debussche A., Vovelle J., Invariant measure of scalar first-order conservation laws with stochastic forcing, arXiv:1310.3779.
- Debussche A., De Moor S., Vovelle J., Diffusion limit for the radiative transfer equation perturbed by a Markovian process, arXiv:1405.2192.
- Debussche A., De Moor S., Vovelle J., Diffusion limit for the radiative transfer equation perturbed by a Wiener process, arXiv:1405.2191.
- Debussche A., De Moor S., Hofmanova M., A regularity result for quasilinear stochastic partial differential equations of parabolic type,arXiv:1401.6369.
- Debussche A. , Hofmanova M., Vovelle J., Degenerate parabolic stochastic partial differential equations: quasilinear case, arXiv:1309.5817.
- De Moor S., Fractional diffusion limit for a stochastic kinetic equation, Stochastic Processes and Applications 124 (2014), no. 3, 1335 –1367.
- Debussche A., Fournier N., Existence of densities for stable-like driven SDE's with Hölder continuous coefficients, Journal of Functional Analysis 264 (2013), no. 8, 1757–1778.
- Dymov A., Statistical mechanics of nonequilibrium systems of rotators with alternated spins, arXiv:1403.1219
- Gawedzki K., Fluctuation relations in stochastic thermodynamics, arXiv:1308.1518.
- Falkovich G., Gawedzki K., Turbulence on hyperbolic plane: the fate of inverse cascade, J. Stat. Phys. 156 (2014), 10–54.
- Hofmanova, M. Degenerate parabolic stochastic partial differential equations, Stochastic Processes and Applications 31 (2013), no. 4, 663–670.
- Hofmanova M., Strong solutions of semilinear stochastic partial differential equations, NoDEA Nonlinear Differential Equations Appl. 20 (2013), no. 3, 757–778.
- Hofmanova M., A Bhatnagar-Gross-Krook approximation to stochastic conservation laws, Ann. Inst. H. Poincare Probab. Statist., to appear.
- Huang G., Kuksin S., KdV equation under periodic boundary conditions and its perturbations, Nonlinearity 27 (2014), 1–28.
- Jaksic V., Nersesyan V., Pillet C.-A., Shirikyan A., Large deviations from a stationary measure for a class of dissipative PDE's with random kicks, hal-00760418 (03/12/2012).
- Jaksic V., Nersesyan V., Pillet C.-A., Shirikyan A., Large deviations and Gallavotti-Cohen principle for dissipative PDE’s with rough noise, hal-00916917 (2013).
- Kleptsyna M., Piatnitskii A., Popier A. Homogenization of random parabolic operators. Diffusion approximation, Stochastic Processes and Applications, sous révision.
- Kleptsyna M., Chiganski P., Cai C., Mixed fractional Brownian motion: the filtering perspective the filtering perspective, Annals of Probability, sous révision.
- Kuksin S.B., Weakly nonlinear stochastic CGL equations, Annales IHP, Prob. Stat. 49 (2013), 1033–1056.
- Kuksin S.B., Neishtadt A.I., On quantum averaging, quantum KAM and quantum diffusion, Russ. Math. Surveys 68 (2013), 335–348.
- Kuksin S.B., Nadirashvili N.S., Analyticity of solutions for quasilinear wave equations and other systems, Proceedings A of the Royal Society of Edinburgh, to appear (2014).
- Kuksin S., Nersesyan V., Stochastic CGL equations without linear dispersion in any space dimension, Stoch PDE: Anal Comp. 1 (2013), no. 3, 389–423.
- Kuksin S., Maiocchi A., Resonant averaging for small solutions of stochastic NLS equations, arXiv:1311.6793
- Le Jan Y., Raimond O., Three examples of Brownian flows on R, Annales de l'IHP, Proba. Stat. (2013), à paraître.
- Martirosyan M., Exponential mixing for the white-forced damped nonlinear wave equation, arXiv:1404.4697.
- Morancey M., Nersesyan V., Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations, hal-00837645 (2013).
- Nardini, C., Gupta S., Ruffo S., Dauxois T., Bouhcet F., Kinetic theory for non-equilibrium stationary states in long-range interacting systems, J. Stat. Mech. (2012), L01002.
- Nardini, C., Gupta S., Ruffo S., Dauxois T., Bouhcet F., Kinetic theory of nonequilibrium stochastic long-range systems: phase transition and bistability, J. Stat. Mech., 12, P12010.
- Nersesyan V., Nersisyan H., Global exact controllability in infinite time of Schrödinger equation: multidimensional case, arXiv:1201.3445 (2012).
- V. Nersesyan, Approximate controllability of Lagrangian trajectories of the 3D Navier-Stokes system by a finite-dimensional force, arXiv:1403.5369 (2014).
- Potters M., Vaillant T., Bouchet F., Sampling microcanonical measures of the 2D Euler equations using Creutz’s algorithm: a phase transition from disorder to order when energy is increased, J. Stat. Mech.: Theory and Experiment (2013), no. 2, P02017.
- Shirikyan A., Zelik S., Exponential attractors for random dynamical systems and applications, Stoch PDE: Anal Comp. 1 (2013), no. 2, 241–281.
- Shirikyan A., Control and mixing for 2D Navier–Stokes equations with space-time localised noise, Ann. Sci. Éc. Norm. Super. 48 (2015), à paraître.
- Thalabard S., Dubrulle B. Bouchet F., Statistical mechanics of the 3D axisymmetric Euler equations in a Taylor-Couette geometry. J. Stat. Mech.: Theory and Experiment (2014), no. 1, P01005.
- Verley G., Chetrite R., Lacoste D., Inequalities generalizing the second law of thermodynamics for transitions between non-stationary states, Phys. Rev. Lett. 108 (2012), 120601.

Publications in proceedings

- Kuksin S.B., Les EDPs hamiltoniennes perturbées et dissipées, Séminaire Laurent Schwartz : EDP et applications (2012-2013), Exp. No. 36, 8p.
- Kuksin S.B., Resonant averaging for weakly nonlinear stochastic
Schrodinger equations, Seminaire Laurent Schwartz : EDP et applications
(2013-2014), Exp. No. 9, 9 p.

- Shirikyan A., Approximate controllability of the viscous Burgers equation on the real line, In: Geometric Control Theory and sub-Riemannian Geometry, G. Stefani, U. Boscain, J.-P. Gauthier, A. Sarychev, M. Sigalotti (Eds.), Springer INdAM Series, Vol. 5; 351–370, 2014.

Conferences

The ANR project STOSYMAP participated in organisation of a number of conferences and workshops.

- The opening conference of the project was held on the 27th of January of 2012 and financed by the Laboratory AGM. The details can be found here.
- Statistique Asymptotique des Processus Stochastiques IX, 11-14 March 2013, University of Le Mans.

- Stochastic and PDE methods in Mathematical Physics, 15-17 September 2014, University of Paris Diderot.
- Statistique Asymptotique des Processus Stochastiques X, 17-20 March 2015, University of Le Mans.

Position

From March of 2013 to August of 2014, Alberto Maiocchi held an 18-month postdoc position shared between the University of Cergy-Pontoise and Institut de Mathématiques de Jussieu.

Abstract of the project

The
aim of this project is to unite efforts of three French teams working
on mathematical aspects of turbulence in various physical media. Past
successes to tackle turbulence mathematically have been scarce and
analytic comprehension has been notoriously difficult. Going further
requires new results in Hamiltonian PDE's, probability theory,
stochastic PDE, a deep qualitative understanding at a physical level,
and possibly insights from numerical simulations. In the last few
years, this type of knowledge was used independently by members of this
project to obtain complementary original results in turbulence
problems. The present joint effort should enable a marked progress in
this important field.

We will consider mathematical issues related to solutions of the 3D (three-dimensional) Navier–Stokes equations (the classical turbulence setup) with high Reynolds numbers, and various statistical characteristics of these solutions. We will also consider simpler related models: the two-dimensional stochastic Navier–Stokes equations (2D turbulence, relevant for meteorology and some fields of physics), the stochastic nonlinear Schrödinger equation in dimensions 1, 2, and 3 (optical turbulence), the Gross–Pitaevskii equation with stochastic perturbations (turbulence in Bose–Einstein condensation), the stochastic Burgers equation (a popular toy model for the classical turbulence), the Korteweg–de Vries equation with small dissipation and random force (another physical model for turbulence in various media).

The project is formed of analytical researches of qualitative properties of solutions for the equations above and for other similar problems. They are supported by numerical studies of the corresponding models. More specifically, our team plans to consider the following questions:

- Ergodicity for 2D Navier–Stokes equations in a bounded domain with stochastic perturbations localized in the physical or Fourier space.
- The inviscid limit of stationary measures in special cases, such as damped/driven linear or completely integrable equations.
- Qualitative study of dispersive equations with various types of stochastic interventions, such as a random dispersion or a random amplitude of a potential.
- Ergodic behavior of the 2D Euler and Navier–Stokes flows and large-scale structures.
- 2D
Navier–Stokes cascades in curved geometry.

In parallel to the above-mentioned questions, we will investigate the more challenging (and unpredictable) problem of qualitative behavior of solutions for the 3D Navier–Stokes system in bounded and unbounded domains. Only few mathematical results related to the phenomenon of turbulence are known in this context, and there is no good understanding of the problem on the physical level of rigor. Our program will include the investigation of following problems for the 3D Navier–Stokes system and other related and/or simplified equations:

- Uniform bounds for the local energy for particular classes of solutions.
- Rigorous results on approximation of physically relevant flows by models with a good understanding of the behavior of solutions.
- Investigation of space-time stationary solutions for the Navier–Stokes system with the Ekman damping and other related PDE's.
- Ergodic
properties of stochastic models of turbulent transport of inertial
particles.

The first aim (and main cost) of this four-year project is to extend the existing research effort, by hiring post-docs to work with the involved researchers. The project will also develop long-term relationships between the partners laboratories, each of them internationally recognized in their own field. Finally, the project will fund workshops and meetings to foster international collaborations and discussions.