Spectral gap for the stochastic primitive equations under degenerate Brownian forcing
Abstract
We prove a weighted Wasserstein spectral gap for the three-dimensional primitive equations on ℰ={u∈ H1∩ L8: ∂zu∈ L4} under finite-rank additive Brownian forcing that is phase-complete finite Fourier and satisfies a quantitative quadratic saturation condition. In particular, four real forcing directions suffice and this number is optimal within the phase-complete class. Consequently, the system is uniquely ergodic and exponentially mixing in the H1 topology. The main difficulties stem from the lack of sufficiently strong moment bounds and the highly degenerate nature of the noise, placing the problem outside the direct reach of standard methods. We introduce a new quantitative stable--compact asymptotic coupling criterion that captures the delicate balance between the available control of the growth of initial perturbations and the accuracy--cost of compensating them by perturbing the noise. For the primitive equations, this leads to a quantitative infinite-dimensional Hörmander analysis requiring tracking the degree and coefficient cost of Lie-bracket polynomials with the Fourier frequency.
Related
- Global well-posedness of the primitive equations with stochastic wind-driven boundary conditions
- Parametric estimation for stochastic wave equations driven by an infinite dimensional fractional Brownian motion
- Pathwise uniqueness for degenerate stochastic differential equations with Hölder continuous coefficients
- Weyl Pseudo Almost Periodic Type Solutions to Semilinear Stochastic Evolution Equations Driven by Fractional Brownian Motion
- Euler Scheme for Stochastic Functional Differential Equations Driven by Fractional Brownian Motion via Fractional Calculus Techniques
BibTeX
@misc{lin2026spectral,
title = {{Spectral gap for the stochastic primitive equations under degenerate Brownian forcing}},
author = {Quyuan Lin and Rongchang Liu and Kening Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05814},
eprint = {2610.05814},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05814},
}
RIS
TY - UNPB
TI - Spectral gap for the stochastic primitive equations under degenerate Brownian forcing
AU - Quyuan Lin
AU - Rongchang Liu
AU - Kening Lu
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05814
UR - https://arxiv.org/abs/2610.05814
AB - We prove a weighted Wasserstein spectral gap for the three-dimensional primitive equations on ℰ={u∈ H^1∩ L^8: ∂_zu∈ L^4} under finite-rank additive Brownian forcing that is phase-complete finite Fourier and satisfies a quantitative quadratic saturation condition. In particular, four real forcing directions suffice and this number is optimal within the phase-complete class. Consequently, the system is uniquely ergodic and exponentially mixing in the H^1 topology. The main difficulties stem from the lack of sufficiently strong moment bounds and the highly degenerate nature of the noise, placing the problem outside the direct reach of standard methods. We introduce a new quantitative stable--compact asymptotic coupling criterion that captures the delicate balance between the available control of the growth of initial perturbations and the accuracy--cost of compensating them by perturbing the noise. For the primitive equations, this leads to a quantitative infinite-dimensional Hörmander analysis requiring tracking the degree and coefficient cost of Lie-bracket polynomials with the Fourier frequency.
ER -