Notes on eventual continuity and ergodicity for SPDEs

TL;DR

This work develops an ergodicity framework for dissipative SPDEs with multiplicative noise by employing eventual continuity and generalized couplings. It links asymptotic stability to a uniform lower bound via $\,\mathrm{Theorem\ 4}\,$ and its $d$-extension $\,\mathrm{Theorem\ 5}\,$, enabling simultaneous existence and uniqueness of invariant measures and weak convergence to them, without assuming prior invariance. The approach is demonstrated on several SPDEs, including the 2D stochastic Navier--Stokes equation, the modified Lagrangian observation process, the 2D hydrostatic Navier--Stokes equation, and the damped Euler--Voigt model, with detailed energy estimates, $d$-eventual continuity, and uniform irreducibility results. The methods rely on Lyapunov structures and generalized couplings to establish eventual regularity and irreducibility, offering broad applicability to multiplicatively forced dissipative SPDEs and enabling robust long-time statistical conclusions. Overall, the notes broaden the toolkit for proving unique ergodicity and asymptotic stability in SPDEs by relaxing strong regularity requirements (e.g., e-property) in favor of eventual continuity, thereby facilitating analysis in nonconventional, multiplicative-noise settings.

Abstract

These notes present an alternative approach to the asymptotic stability of stochastic partial differential equations driven by multiplicative noise, applicable to a wide range of dissipative systems. The method builds on general criteria established in \cite{GLLL2024b,L2023}, utilizing the eventual continuity and generalized coupling techniques.

Theorems & Definitions (54)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• Definition 2.7
• Theorem 2.8
• Remark 2.9
• proof : Proof of Theorem \ref{['Thm 5']}