Sharp quasi-invariance threshold for the cubic Szeg\H{o} equation

TL;DR

This paper investigates how Gaussian measures evolve under the cubic Szeg"H{o} equation, revealing a sharp transition at a critical regularity level where measures switch from being quasi-invariant to mutually singular.

Contribution

It establishes the first known transition point for Gaussian measure behavior under a Hamiltonian PDE flow, specifically at s=1 for the cubic Szeg"H{o} equation.

Findings

For s>1, the measure is quasi-invariant under the flow.

For s<1, the measure and initial measure are mutually singular.

The transition occurs sharply at s=1, marking a phase change in measure behavior.

Abstract

We consider the 1-dimensional cubic Szeg\H{o} equation with data distributed according to the Gaussian measure with inverse covariance operator $(1-\partial_x^2)^\frac s2$, where $s>\frac12$. We show that, for $s>1$, this measure is quasi-invariant under the flow of the equation, while for $s<1$, $s\neq \frac34$, the transported measure and the initial Gaussian measure are mutually singular for almost every time. This is the first observation of a transition from quasi-invariance to singularity in the context of the transport of Gaussian measures under the flow of Hamiltonian PDEs.