On invariant Gibbs measures for the generalized KdV equations

TL;DR

This paper constructs global solutions for the defocusing generalized KdV equations on the circle with initial data distributed by the Gibbs measure, demonstrating invariance of the measure over time using advanced probabilistic techniques.

Contribution

It introduces a method to handle high-degree nonlinearities in generalized KdV equations using Hermite polynomials and white noise functionals, establishing measure invariance.

Findings

Global-in-time solutions with Gibbs measure initial data

Invariance of the Gibbs measure under the flow

Handling arbitrary high-degree nonlinearities

Abstract

We consider the defocusing generalized KdV equations on the circle. In particular, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure. In handling a nonlinearity of an arbitrary high degree, we make use of the Hermite polynomials and the white noise functional.