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We study the Benjamin–Ono equation, posed on the torus. We prove that an infinite sequence of weighted Gaussian measures, constructed in our previous work, are invariant by the flow of the equation. These measures are supported by Sobolev spaces of increasing regularity. As a by-product, we deduce informations on the long-time behavior of regular solutions. To our knowledge, this is the first result which gives an evidence about recurrence properties of the Benjamin–Ono equation flow.

Invariant Measures and Long-Time Behavior for the Benjamin–Ono Equation