  In this paper, we study the Gibbs measures for periodic generalized
Korteweg-de Vries equations (gKdV) with quartic or higher nonlinearities. In
order to bypass the analytical ill-posedness of the equation in the Sobolev
support of the Gibbs measures, we establish deterministic well-posedness of the
gauged gKdV equations within the framework of the Fourier-Lebesgue spaces. Our
argument relies on bilinear and trilinear Strichartz estimates adapted to the
Fourier-Lebesgue setting. Then, following Bourgain's invariant measure
argument, we construct almost sure global-in-time dynamics and show invariance
of the Gibbs measures for the gauged equations. These results can be brought
back to the ungauged side by inverting the gauge transformation and exploiting
the invariance of the Gibbs measures under spatial translations. We thus
complete the program initiated by Bourgain (1994) on the invariance of the
Gibbs measures for periodic gKdV equations.
