We consider the weakly damped stochastic nonlinear Schr\"odinger (NLS) equation on the real line, driven by a noise that is white in time and smooth in space. Assuming that the noise is sufficiently non-degenerate, we prove that the equation has a unique stationary measure in the class of probability measures concentrated on $H^2$, and establish polynomial mixing in the dual-Lipschitz metric over $H^1$. We do not impose any restriction on the size of the damping. The proof is based on a coupling argument, whose key ingredient is a Foia\c{s}-Prodi-type estimate in the $H^1$-norm, derived by means of a Lyapunov functional adapted to the linearized NLS dynamics. To compensate for the loss of compactness, we combine this estimate with a truncated Poincar\'e inequality and a space-time weight function quantifying the spatial decay of solutions.
We prove polynomial mixing for the defocusing damped cubic stochastic nonlinear Schr\"odinger equation on the three-dimensional torus under saturating smooth finite rank Brownian forcing. The mixing rate is measured in the $p$-Wasserstein metric induced by the $H^1$ distance for every $1\le p<\infty$. We also obtain sh...
We establish a weighted Wasserstein spectral gap for the three-dimensional damped cubic wave equation with genuinely finite rank Brownian forcing. Under a saturation condition, the gap holds with respect to the negative phase topology $\mathcal E_s=H^{-s}\times H^{-1-s}$ for every $0<s<1/2$, from which we deduce unique...
To quantitatively characterize the long-time dynamics of the periodic damped stochastic Korteweg--de Vries (sKdV) equation driven by additive noise, we investigate the uniform-in-time error estimates for a Lie--Trotter operator splitting approximation. This splitting combines the exact deterministic KdV flow with the e...
We prove a weighted Wasserstein spectral gap for the three-dimensional primitive equations on $\mathscr E=\{u\in H^1\cap L^8: \partial_zu\in L^4\}$ under finite-rank additive Brownian forcing that is phase-complete finite Fourier and satisfies a quantitative quadratic saturation condition. In particular, four real forc...
For each positive integer $N$, we construct global spectral group actions for the defocusing and focusing nonlinear Schr\"odinger hierarchies on initial-data classes containing $W^{2N+1,\infty}(\mathbb R)$. For $N\ge2$, the quadratic flow gives global classical cubic NLS solutions. The construction extends the Sato--Se...
S. Kotani, Yi-Qian Wang, Jia-Hao Xu et al.· 1 citation
This paper studies a finite-horizon mixed $H_2/H_\infty$ feedback Nash game for stochastic evolution equations on a separable Hilbert space. The drift generator is unbounded, the remaining coefficients are bounded, and the one-dimensional Brownian diffusion depends on the state, control, and disturbance. The $H_2$ chan...
Ming-Yang Shen, Wei-Hai Zhang, Qing-Xin Meng et al.· 0 citations
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