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Invariant Gibbs measures and global strong solutions for nonlinear Schr\"odinger equations in dimension two
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abstract
We consider the defocusing nonlinear Schr\"odinger 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.
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$\Phi^4_3$ Theory from many-body quantum Gibbs states
The Gibbs state of an interacting Bose gas on the three-dimensional torus is proven to converge, in a tuned semiclassical limit, to the renormalized Phi^4_3 measure.
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