Invariance of the Gibbs measure for the periodic quartic gKdV

Abstract

We prove invariance of the Gibbs measure for the (gauge transformed) periodic quartic gKdV. The Gibbs measure is supported on $H^s(\mathbb{T})$ for $s<\frac{1}{2}$, 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 with 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 $H^{\frac{1}{2}}(\mathbb{T})$.