10 papers
Exponential mixing for the stochastic Navier--Stokes equation with localized noise
Ziyu Liu, Shengquan Xiang, Zhifei Zhang
This paper studies the 2D Navier--Stokes equation on a bounded domain with Navier-slip boundary conditions, driven by spatially localized stationary forcing generated by a stochast…
Exponential mixing for Korteweg-de Vries equation with localized noise
Yuxuan Chen, Shengquan Xiang, Zhifei Zhang +1
We establish exponential mixing for the randomly forced and weakly damped KdV equation in . The noise is bounded, localized, and degenerate in high frequencies. Ou…
Asymptotic strong Feller and weak observability inequality
Ziyu Liu, Shengquan Xiang
For a class of non-autonomous linear SPDEs, we establish the equivalence among asymptotic regularization, weak observability, and approximate null controllability for the associate…
A note on the strong Feller property via the moment method
Ziyu Liu, Shengquan Xiang
This note studies the 1D stochastic heat equation driven by a one-dimensional Brownian motion. We prove that the associated Markov process satisfies the strong Feller property unde…
Exponential mixing for the stochastic Allen--Cahn equation with localized white noise
Ziyu Liu, Shengquan Xiang, Zhifei Zhang
This paper studies the 1D stochastic Allen--Cahn equation on a bounded domain driven by localized white noise. We prove that the associated Markov process admits a unique invariant…
Exponential mixing for nonlinear Schrödinger equations perturbed by bounded degenerate noise
Yuxuan Chen, Shengquan Xiang, Zhifei Zhang
We prove the exponential convergence to a unique invariant measure for locally damped nonlinear Schrödinger equations, perturbed by bounded noise acting on only two Fourier modes.…
Donsker-Varadhan large deviation principle for locally damped and randomly forced NLS equations
Yuxuan Chen, Shengquan Xiang
We study large deviations from the invariant measure for nonlinear Schrödinger equations with colored noises on determining modes. The proof is based on a new abstract criterion, i…
Exponential mixing for the randomly forced NLS equation
Yuxuan Chen, Shengquan Xiang, Zhifei Zhang +1
This paper investigates exponential mixing of the invariant measure for randomly forced nonlinear Schrödinger equation, with damping and random noise localized in space. Our study…
Local large deviations for randomly forced nonlinear wave equations with localized damping
Yuxuan Chen, Ziyu Liu, Shengquan Xiang +1
We study the large deviation principle (LDP) for locally damped nonlinear wave equations perturbed by a bounded noise. When the noise is sufficiently non-degenerate, we establish t…
Exponential mixing for random nonlinear wave equations: weak dissipation and localized control
Ziyu Liu, Dongyi Wei, Shengquan Xiang +2
We establish a new criterion for exponential mixing of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equation…