A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations

Tadahiro Oh (1) , Laurent Thomann (2)

1 Edin. - University of Edinburgh
2 IECL - Institut Élie Cartan de Lorraine

We consider the defocusing nonlinear Schr\"odinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in $\R^2$. Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure.

Mots clés: Hermite polynomial; white noise functional; Gibbs measure; Laguerre polynomial; nonlinear Schrödinger equation; Wick ordering

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Tadahiro Oh, Laurent Thomann. A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations. Stochastics and Partial Differential Equations: Analysis and Computations, 2018, 6 (3), pp.397--445. ⟨10.1007/s40072-018-0112-2⟩. ⟨hal-01194713v2⟩