Quasi invariant Gaussian measures for the nonlinear Schr\"odinger equation on $\mathbb T^2$

TL;DR

This paper proves that Gaussian measures with certain inverse covariance properties are quasi-invariant under the flow of the 2D nonlinear Schrödinger equation on the torus, with the Radon-Nikodym derivative in all local L^p spaces.

Contribution

It introduces a new abstract quasi-invariance argument and extends the understanding of measure transport for the 2D nonlinear Schrödinger equation.

Findings

Gaussian measures are quasi-invariant for s>2

Radon-Nikodym density belongs to all local L^p spaces

New probabilistic and space-time estimate techniques

Abstract

We study the transport of Gaussian measures under the flow of the 2-dimensional defocusing Schr\"odinger equation $i \partial_t u + \Delta u = |u|^{2k} u$ posed on $\mathbb T^2$. In particular, we show that the Gaussian measures with inverse covariance $\|u\|_{H^s}^2$, are quasi-invariant under the flow for $s>2$. Moreover, we show that the Radon-Nykodim density belongs to every $L^p$ space, locally in space. The proof relies on the physical-space energies introduced in [52], as well as a new abstract quasi-invariance argument that allows us to combine space-time estimates, along the flow with probabilistic bounds on the support of the measure.