Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS

TL;DR

This paper introduces a Wiener-based randomization technique on unbounded domains that enhances Strichartz estimates for Schrödinger equations, leading to almost sure local well-posedness results for energy-critical NLS below the energy space.

Contribution

It develops a novel Wiener randomization method on unbounded domains linked to modulation spaces, improving well-posedness results for nonlinear Schrödinger equations.

Findings

Enhanced Strichartz estimates for randomized data

Almost sure local well-posedness of energy-critical NLS below energy space

New randomization technique based on Wiener decomposition

Abstract

We consider a randomization of a function on $\mathbb R^d$ that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schr\"odinger equation. As an example, we also show that the energy-critical cubic nonlinear Schr\"odinger equation on $\mathbb R^4$ is almost surely locally well-posed with respect to randomized initial data below the energy space.