Transport of gaussian measures by the flow of the nonlinear Schr\"odinger equation

TL;DR

This paper establishes quasi-invariance of Gaussian measures under the flow of the 1D quintic Schrödinger equation, demonstrating new smoothing properties and extending results to odd power nonlinearities, with global results in the defocusing case.

Contribution

It introduces a new smoothing property for solutions and proves quasi-invariance of Gaussian measures for the 1D quintic Schrödinger equation, including extensions to odd nonlinearities.

Findings

Gaussian measures are quasi-invariant under the flow

Global quasi-invariance in the defocusing case

Local quasi-invariance in the focusing case due to blow-up

Abstract

We prove a new smoothing type property for solutions of the 1d quintic Schr\"odinger equation. As a consequence, we prove that a family of natural gaussian measures are quasi-invariant under the flow of this equation. In the defocusing case, we prove global in time quasi-invariance while in the focusing case because of a blow-up obstruction we only get local in time quasi-invariance. Our results extend as well to generic odd power nonlinearities.