Ergodic and mixing properties of the 2D Navier–Stokes equations with a degenerate multiplicative Gaussian noise
Abstract
In this paper, we establish the ergodic and mixing properties of stochastic 2D Navier–Stokes equations driven by a highly degenerate multiplicative Gaussian noise. The noise can appear in as few as four directions, and its intensity depends on the solution. The case of additive Gaussian noise was previously treated by Hairer and Mattingly (Ann. of Math. (2) 164 (2006) 993–1032). To derive the ergodic and mixing properties in the present setting, we employ Malliavin calculus to establish the asymptotically strong Feller property. The primary challenge lies in proving the “invertibility” of the Malliavin matrix, which differs fundamentally from the additive case.
Fields of Study & Subjects
Stochastic processes and financial applications
Stability and Controllability of Differential Equations
Navier-Stokes equation solutions
Publication details
- Journal / source
- The Annals of Applied Probability
- Publisher
- Institute of Mathematical Statistics
- Volume
- 36
- Issue
- 2
- Pages
- 1562-1614
- Publication date
- 1 April 2026