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arXiv · 2404.14950

Sharp quasi-invariance threshold for the cubic Szegő equation

James Coe·Leonardo Tolomeo
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Abstract

We consider the 1-dimensional cubic Szegő equation with data distributed according to the Gaussian measure with inverse covariance operator $(1-\partial_x^2)^\frac s2$, 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.

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James Coe, Leonardo Tolomeo. 2024-04-23. Sharp quasi-invariance threshold for the cubic Szegő equation. https://doi.org/10.2140/apde.2026.19.1107

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