Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schr\"odinger equation

TL;DR

This paper proves that Gaussian measures on certain Sobolev spaces are quasi-invariant under the flow of the cubic fourth order nonlinear Schrödinger equation on the circle, advancing understanding of measure dynamics in high-order PDEs.

Contribution

It establishes quasi-invariance of Gaussian measures for the cubic fourth order NLS, a novel result for high-order nonlinear Schrödinger equations.

Findings

Gaussian measures are quasi-invariant for s > 3/4

Results apply to Sobolev spaces on the circle

Advances measure-theoretic understanding of high-order NLS

Abstract

We consider the cubic fourth order nonlinear Schr\"odinger equation on the circle. In particular, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac34$, are quasi-invariant under the flow.