Sharp quasi-invariance threshold for the cubic Szegő equation

Coe, James; Tolomeo, Leonardo

doi:10.2140/apde.2026.19.1107

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Sharp quasi-invariance threshold for the cubic Szegő equation

James Coe and Leonardo Tolomeo

Vol. 19 (2026), No. 6, 1107–1164

DOI: 10.2140/apde.2026.19.1107

Abstract

We consider the 1-dimensional cubic Szegő equation with data distributed according to the Gaussian measure with inverse covariance operator ${(1 - \partial_{x}^{2})}^{s}$ , where $s > \frac{1}{2}$ . We show that, for $s > 1$ , this measure is quasi-invariant under the flow of the equation, while for $s < 1$ , $s \neq \frac{3}{4}$ , 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.

Keywords

Szegő equation, quasi-invariance, phase transition

Mathematical Subject Classification

Primary: 35R60, 37A40

Secondary: 60H30

Milestones

Received: 23 April 2024

Revised: 23 June 2025

Accepted: 19 September 2025

Published: 13 July 2026

Authors

James Coe
School of Mathematics The University of Edinburgh Edinburgh United Kingdom

Leonardo Tolomeo
School of Mathematics The University of Edinburgh Edinburgh United Kingdom

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Vol. 19, No. 6, 2026