Invariance of the Gibbs measures for periodic generalized Korteweg-de Vries equations

In this paper, we study the Gibbs measures for periodic generalized Korteweg-de Vries equations (gKdV) with quartic or higher nonlinearities. order to bypass analytical ill-posedness of equation in Sobolev support measures, establish deterministic well-posedness gauged gKdV within framework Fourier-Lebesgue spaces. Our argument relies on bilinear and trilinear Strichartz estimates adapted setting. Then, following Bourgain’s invariant measure argument, construct almost sure global-in-time dynamics show invariance equations. These results can be brought back ungauged side by inverting gauge transformation exploiting under spatial translations. We thus complete program initiated Bourgain [Comm. Math. Phys. 166 (1994), pp 1–26]