Invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations

Tadahiro Oh (1) , Laurent Thomann (2)

1 Edin. - University of Edinburgh
2 EDP - Equations aux dérivées partielles

We consider the defocusing nonlinear Schrödinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in $\mathbb{R}^2$. In particular, we discuss 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: nonlinear Schrödinger equation; Laguerre polynomial; white noise functional; Gibbs measure; Wick ordering; Hermite polynomial

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

Tadahiro Oh, Laurent Thomann. Invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations. 2015. ⟨hal-01194713v1⟩