Invariance of the Gibbs measure for the Benjamin–Ono equation

Abstract

In this paper we consider the periodic Benjemin-Ono equation.We establish the invariance of the Gibbs measure associated to this equation, thus answering a question raised in Tzvetkov [28]. As an intermediate step, we also obtain a local well-posedness result in Besov-type spaces rougher than $L^2$, extending the $L^2$ well-posedness result of Molinet [20].