Sharp quasi-invariance threshold for the cubic Szegő equation
Abstract
We consider the 1-dimensional cubic Szegő equation with data distributed according to the Gaussian measure with inverse covariance operator (1 -∂ 2x ) s , where s > 1 2 .We show that, for s > 1, this measure is quasi-invariant under the flow of the equation, while for s < 1, s ̸ = 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.
Fields of Study & Subjects
Advanced Mathematical Physics Problems
Nonlinear Waves and Solitons
Geometry and complex manifolds
Publication details
- Journal / source
- Analysis & PDE
- Volume
- 19
- Issue
- 6
- Pages
- 1107–1164
- Publication date
- 13 July 2026
- Language
- EN