Theoretical analysis demonstrates quasi-invariance of Gaussian measures in the periodic Benjamin–Ono–BBM equation, indicating persistent statistical stability across all globally well-posed regimes.
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.
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Justin Forlano (2026) studied this question.
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