Low-regularity invariant measure for the complex-valued mKdV

TL;DR

This paper constructs a low-regularity invariant measure for the complex-valued, twice-renormalized mKdV equation on the torus, extending previous work to more complex and less regular function spaces.

Contribution

It introduces the first invariant measure supported at low-regularity for the complex-valued mKdV, utilizing Fourier-Lebesgue spaces due to the equation's complexity.

Findings

Constructed a low-regularity invariant measure for complex mKdV

Extended invariant measure theory to Fourier-Lebesgue spaces

Addressed complexities of complex-valued versus real-valued mKdV

Abstract

In this paper we consider the twice-renormalized, complex-valued modified KdV (mKdV) on the one-dimensional torus introduced by Chapouto. Our main result is the construction of an invariant measure supported at low-regularity. This work complements the work of Kenig et al., which constructed invariant measures supported in higher-regularity spaces for the non-renormalized mKdV. Due to the low-regularity of the support of the measure, we are forced to work in Fourier-Lebesgue spaces. The fact that we consider the complex-valued mKdV makes the problem more complicated than the real-valued case, which was previously considered.