  We prove invariance of the Gibbs measure for the (gauge transformed) periodic
quartic gKdV. The Gibbs measure is supported on H^s for s<1/2, and the quartic
gKdV is analytically ill-posed in this range. In order to consider the flow in
the support of the Gibbs measure, we combine a probabilistic argument and the
second iteration and construct local-in-time solutions to the (gauge
transformed) quartic gKdV almost surely in the support of the Gibbs measure.
Then, we use Bourgain's idea to extend these local solutions to global
solutions, and prove the invariance of the Gibbs measure under the flow.
Finally, Inverting the gauge, we construct almost sure global solutions to the
(ungauged) quartic gKdV below H^{1/2}.
