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.

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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