Invariant Gibbs measures and global strong solutions for nonlinear Schrödinger equations in dimension two

Abstract

We consider the defocusing nonlinear Schrödinger equation on ${\mathbb{T}}^2$ with Wick ordered power nonlinearity, and prove almost sure global well-posedness with respect to the associated Gibbs measure. The heart of the matter is the uniqueness of the solution as limit of solutions to canonically truncated systems. The invariance of the Gibbs measure under the global dynamics follows as a consequence. The proof relies on the novel idea of random averaging operators.

61 pages. [V2] Final version. To appear in Ann. of Math

References in corpus (3)

• Invariant Gibbs Measures and a.s. Global Well-Posedness for Coupled KdV Systems
• Singular stochastic PDEs
• Probabilistic local well-posedness of the cubic nonlinear wave equation in negative Sobolev spaces

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