Context of the project:

There has been in the last thirteen years considerable progresses in our understanding of some classes of random partial differential equations (PDEs) after the 2008 work of Burq & Tzvetkov on supercritical wave equations with random initial conditions and the 2014 works of Hairer and Gubinelli, Imkeller & Perkowski on singular stochastic elliptic and parabolic PDEs. These fundamental developments have laid the basis of a robust solution theory for equations that are classically ill-posed, due to a low regularity random data in the problem. While the regularity structures and paracontrolled approaches to stochastic PDEs have their roots in T. Lyons’ theory of rough paths, the progresses in the dispersive side have their roots in Bourgain’s works on invariant Gibbs measures for the nonlinear Schr ̈odinger equation in the mid nineties. In recent years, it became obvious that some of the conceptual ideas introduced in in the dispersive setting echo the strategy adopted to study singular stochastic PDEs. For instance, in both situations, any potential solution is decomposed as the sum of a ‘finite dimensional’ singular part enjoying regularity properties beyond the deterministic analysis, and a remainder part that is more regular and can be treated using purely analytical tools adapted to the equation – Bourgain spaces, regularity or paracontrolled structures. This project aims at a deeper clarification of the common features and key differences between the analysis of dispersive PDEs with random initial data and the analysis of singular stochastic PDEs. The objective will also consist in identifying new possible interactions between these do- mains, as well as developing specific tools for these classes of equations. The project will build upon ongoing collaborations between members of the dispersive and singular stochastic PDEs communities, and it will undoubtedly foster new ones.

Team members:

I. Bailleul (P.I.), N. Burq, V. Dang, A. Deya, C. Labbe, T. Robert, L. Thomann, N. Tzvetkov (co-P.I.), C. Sun L. Zambotti,

Activity:

Octobre 2023 Meeting in Nancy

May 2024 Meeting in Brest: Talks by Alberto, Alexis, Jiasheng, Aurelien, Van Duong, Yueh-Sheng, Cyril, Jacob, Leonard, Yvain and Laurent.

Publications:

Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space
A. Debussche, R. Liu, N. Tzvetkov, N. Visciglia

New non-degenerate invariant measures for the Benjamin-Ono equation
N. Tzvetkov

Blow-up for the 1D cubic NLS
V. Banica, R. Lucà, N. Tzvetkov, L. Vega

Φ43 measures on compact Riemannian 3-manifolds
I. Bailleul, N.V. Dang, L. Ferdinand, T.D. Tô

Bilinear Strichartz estimates and almost sure global solutions for the nonlinear Schrödinger equation
N. Burq, A. Poiret, L. Thomann

Mean field singular stochastic PDEs
I. Bailleul, N. Moench

Dynamics of quintic nonlinear Schrödinger equations in H2/5+(𝕋)
J. Bernier, B. Grébert, T. Robert

Uniqueness of the Φ43 measures on closed Riemannian 3-manifolds
I. Bailleul

Semi-classical observation suffices for observability: wave and Schrödinger equations
N. Burq, B. Dehman, J. Le Rousseau

Post-Lie algebras of derivations and regularity structures
J.-D. Jacques, L. Zambotti

Measure propagation along 𝒞0-vector field and wave controllability on a rough compact manifold
N. Burq, B. Dehman, J. Le Rousseau

Wilson-Itô diffusions
I. Bailleul, I. Cheryvev, M. Gubinelli

Quasi-invariance of Gaussian measures for the 3d energy critical nonlinear Schrödinger equation
C. Sun, N. Tzvetkov

A remark on randomization of a general function of negative regularity
T. Oh, M. Okamoto, O. Pocovnicu, N. Tzvetkov

Domains of dependence for subelliptic wave equations and unique continuation for fractional powers of Hörmander's operators
N. Burq, C. Zuily

Quantitative observability for one-dimensional Schrödinger equations with potentials
P. Su, C. Sun, X. Yuan

On the 1d stochastic Schrödinger product
A. Deya

Random models on regularity-integrability structures
I. Bailleul, M. Hoshino

Existence, uniqueness, and universality of global dynamics for the fractional hyperbolic Φ43-mode
R. Liu, N. Tzvetkov, Y. Wang

Spectrally cut-off GFF, regularized Φ4 measure, and reflection positivity
I. Bailleul, N.V. Dang, L. Ferdinand, G. Leclerc, J. Lin

Anderson stochastic quantization equation
H. Eurly, A. Mouzard, T. Robert

Construction and spectrum of the Anderson Hamiltonian with white noise potential on R2 and R3
Y.-S. Hsu, C. Labbé

Cutoff phenomenon in nonlinear recombinations
P. Caputo, C. Labbé

On the stability of the Abrikosov lattice in the Lowest Landau Level
P. Germain, V. Schwinte, L. Thomann

Convergence of dynamical stationary fluctuations
C. Labbé, B. Laslier, F. Toninelli, L. Zambotti

Non-local quasilinear singular SPDEs
I. Bailleul, H. Eulry

Probabilistic well-posedeness for the nonlinear Schrödinger equation on the 2d sphere I: positive regularities
N. Burq, N. Camps, Ch. Sun, N. Tzvetkov

The Second Picard iteration of NLS on the 2d sphere does not regularize Gaussian random initial data
N. Burq, N. Camps, M. Latocca, Ch. Sun, N. Tzvetkov

Almost surely smoothed scattering for cubic NLS
N. Burq, H. Koch, N. Visciglia, N. Tzvetkov

Measure and continuous vector field at a boundary I: propagation equations and wave observability
N. Burq, B. Dehman, J. Le Rousseau

Measure and continuous vector field at a boundary II: geodesics and support propagation
N. Burq, B. Dehman, J. Le Rousseau

The Cauchy problem for the periodic Kadomtsev-Petviashvili-II equation below L2
S. Herr, R. Schippa, N. Tzvetkov

Delocalized eigenvectors of transitive graphs and beyond
N. Burq, C. Letrouit

New bounds on the high Sobolev norms of the 1d NLS solutions
D. Berti, F. Planchon, N. Tzvetkov, N. Visciglia

Renormalization of random models: a review
I. Bailleul, M. Hoshino