Improved quasi-invariance result for the periodic Benjamin-Ono-BBM equation
Justin Forlano
TL;DR
This work proves a refined quasi-invariance result for the periodic BO-BBM equation by extending Gaussian measure quasi-invariance from the previously known high-regularity range to the full well-posedness band $s>\tfrac{1}{2}$. The authors blend the probabilistic framework of Coe–Tolomeo with an iteration strategy inspired by Forlano–Tolomeo to obtain long-time, high-integrability bounds for the transported density on energy-localised sets $B_R$, by establishing uniform exponential integrability of an energy increment functional $\mathcal{Q}_{s,N}$ and constructing short-time $L^p$ bounds that survive iteration. The analysis hinges on a careful decomposition of $\mathcal{Q}_{s,N}$ into $\mathcal{Q}^{(1)}_{s,N}$ and $\mathcal{Q}^{(2)}_{s,N}$ and a detailed treatment of random oscillations arising from the Gaussian initial data, enabling uniform-in-$N$ control of the density after localisation. The paper also shows a lower-bound mechanism for $0<s\le\tfrac{1}{2}$, suggesting a threshold for quasi-invariance and clarifying the role of the borderline dispersion in this model. Overall, the results advance understanding of how dispersion strength interacts with probabilistic invariance properties in nearly critical Hamiltonian PDEs and set a framework for achieving long-time probabilistic well-posedness insights via localisation and variational techniques.
Abstract
We extend recent results of Genovese-Luca-Tzvetkov (2022) regarding the quasi-invariance of Gaussian measures under the flow of the periodic Benjamin-Ono-BBM (BO-BBM) equation to the full range where BO-BBM is globally well-posed. The main difficulty is due to the critical nature of the dispersion which we overcome by combining the approach of Coe-Tolomeo (2024) with an iteration argument due to Forlano-Tolomeo (2024) to obtain long-time higher integrability bounds on the transported density.