  The Benjamin-Ono equation describes the propagation of internal waves in a
stratified fluid. In the present work, we study large time dynamics of its
regular solutions via some probabilistic point of view. We prove the existence
of an invariant measure concentrated on $C^\infty(\T)$ and establish some
qualitative properties of this measure. We then deduce a recurrence property of
regular solutions and other corollaries using ergodic theorems. The approach
used in this paper applies to other equations with infinitely many conservation
laws, such as the KdV and cubic Schr\"odinger equations in 1D. It uses the
fluctuation-dissipation-limit approach and relies on a \textit{uniform}
smoothing lemma for stationary solutions to the damped-driven Benjamin-Ono
equation.
