Exponential mixing for the stochastic Allen--Cahn equation with localized white noise
Probability
2026-05-08 v1 Analysis of PDEs
Optimization and Control
Abstract
This paper studies the 1D stochastic Allen--Cahn equation on a bounded domain driven by localized white noise. We prove that the associated Markov process admits a unique invariant measure and is exponential mixing. The main challenge lies in the interaction between localized nature of the noise and non-trivial global dynamics of the system. To overcome this, our approach relies on two ingredients from PDE control theory: stabilization for the linearized system and global steady-state controllability for the nonlinear equation. The stabilization result is derived using the weak observability and Fenchel--Rockafellar duality, while the global controllability relies on quasi-static deformations combined with global dynamics.
Cite
@article{arxiv.2605.06009,
title = {Exponential mixing for the stochastic Allen--Cahn equation with localized white noise},
author = {Ziyu Liu and Shengquan Xiang and Zhifei Zhang},
journal= {arXiv preprint arXiv:2605.06009},
year = {2026}
}