On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations

Abstract

We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails.

References in corpus (7)

• Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations
• Local Martingale and Pathwise Solutions for an Abstract Fluids Model
• Global Existence and Regularity for the 3D Stochastic Primitive Equations of the Ocean and Atmosphere with Multiplicative White Noise
• Existence and Regularity of Invariant Measures for the Three Dimensional Stochastic Primitive Equations
• On ergodicity of some Markov processes
• Unique Ergodicity for Fractionally Dissipated, Stochastically Forced 2D Euler Equations
• Self-truncation and scaling in Euler-Voigt-$\alpha$ and related fluid models

Cited by in corpus (7)

• The small-mass limit and white-noise limit of an infinite dimensional Generalized Langevin Equation
• Couplings via comparison principle and exponential ergodicity of SPDEs in the hypoelliptic setting
• Exponential ergodicity for a stochastic two-layer quasi-geostrophic model
• Statistical Properties of 2D Stochastic Navier-Stokes Equations with Time-Periodic Forcing and Degenerate Stochastic Forcing
• Ergodicity for stochastic equation of Navier--Stokes type
• Generalized couplings and convergence of transition probabilities
• The invariant measure of a walking droplet in hydrodynamic pilot-wave theory