Exponential mixing for Korteweg-de Vries equation with localized noise

Yuxuan Chen; Shengquan Xiang; Zhifei Zhang; Jia-Cheng Zhao

Exponential mixing for Korteweg-de Vries equation with localized noise

Analysis of PDEs 2026-08-03 v1 Dynamical Systems Optimization and Control Probability

Authors: Yuxuan Chen , Shengquan Xiang , Zhifei Zhang , Jia-Cheng Zhao

Abstract

We establish exponential mixing for the randomly forced and weakly damped KdV equation in $L^2(\mathbb{T})$ . The noise is bounded, localized, and degenerate in high frequencies. Our proof relies on a general probabilistic framework in [11,33], nonlinear smoothing for KdV and its linearization via normal form transformation, and stabilization of the system by localized force. This paper continues a series of works connecting asymptotic compactness, control theory, and ergodicity and mixing for randomly forced dispersive PDEs.

Keywords

exponential mixing KdV equation localized noise random forcing weak damping ergodicity nonlinear smoothing

Cite

@article{arxiv.2608.01863,
  title  = {Exponential mixing for Korteweg-de Vries equation with localized noise},
  author = {Yuxuan Chen and Shengquan Xiang and Zhifei Zhang and Jia-Cheng Zhao},
  journal= {arXiv preprint arXiv:2608.01863},
  year   = {2026}
}