# Almost sure global well-posedness for the energy supercritical Schr\"odinger equations

## TL;DR

This paper demonstrates almost sure global well-posedness for energy supercritical Schrödinger equations on the three-dimensional torus by constructing invariant measures and analyzing their properties, with potential extensions to other dimensions.

## Contribution

It introduces a novel approach combining fluctuation-dissipation techniques and Gibbs measure theory to establish global solutions in supercritical regimes.

## Key findings

- Constructed invariant probability measures supported on Sobolev spaces.
- Proved global well-posedness almost surely on the measure supports.
- Showed measures are invariant and solutions are recurrent in time.

## Abstract

We consider the Schr\"odinger equations with arbitrary (large) power non-linearity on the three-dimensional torus. We construct non-trivial probability measures supported on Sobolev spaces and show that the equations are globally well-posed on the supports of these measures, respectively. Moreover, these measures are invariant under the flows that are constructed. Therefore, the constructed solutions are recurrent in time.\\ Also, we show \textit{slow growth} control on the time evolution of the solutions. A generalization to any dimension is given. Our proof relies on a new approach combining the fluctuation-dissipation method and some features of the Gibbs measures theory for Hamiltonian PDEs. The strategy of the paper applies to other contexts